Abrasive article including shaped abrasive particles

ABSTRACT

A shaped abrasive particle including a body having a first major surface, a second major surface, and a side surface extending between the first major surface and the second major surface, wherein the body includes a sharpness-shape-strength factor (3SF) within a range between about 0.7 and about 1.7 and a Shape Index within a range between at least about 0.51 and not greater than about 0.99.

CROSS-REFERENCE TO RELATED APPLICATION(S)

This application claims priority under 35 U.S.C. §119(e) to U.S. PatentApplication No. 61/979,316, filed Apr. 14, 2014, U.S. Patent ApplicationNo. 62/079,218, filed Nov. 13, 2014, and U.S. Patent Application No.62/106,545, filed Jan. 22, 2015, entitled “ABRASIVE ARTICLE INCLUDINGSHAPED ABRASIVE PARTICLES,” by, Kristin BREDER, Jennifer H. CZEREPINSKI,Flavien FREMY, David LOUAPRE, Samuel S. MARLIN, Yves BOUSSANT-ROUX, andSujatha IYENGAR, which is assigned to the current assignee hereof andincorporated herein by reference in its entirety.

BACKGROUND Field of the Disclosure

The following is directed to abrasive articles, and particularly,abrasive articles including shaped abrasive particles.

Description of the Related Art

Abrasive particles and abrasive articles made from abrasive particlesare useful for various material removal operations including grinding,finishing, and polishing. Depending upon the type of abrasive material,such abrasive particles can be useful in shaping or grinding a widevariety of materials and surfaces in the manufacturing of goods. Certaintypes of abrasive particles have been formulated to date that haveparticular geometries, such as triangular shaped abrasive particles andabrasive articles incorporating such objects. See, for example, U.S.Pat. Nos. 5,201,916; 5,366,523; and 5,984,988.

Three basic technologies that have been employed to produce abrasiveparticles having a specified shape are (1) fusion, (2) sintering, and(3) chemical ceramic. In the fusion process, abrasive particles can beshaped by a chill roll, the face of which may or may not be engraved, amold into which molten material is poured, or a heat sink materialimmersed in an aluminum oxide melt. See, for example, U.S. Pat. No.3,377,660 (disclosing a process including flowing molten abrasivematerial from a furnace onto a cool rotating casting cylinder, rapidlysolidifying the material to form a thin semisolid curved sheet,densifying the semisolid material with a pressure roll, and thenpartially fracturing the strip of semisolid material by reversing itscurvature by pulling it away from the cylinder with a rapidly drivencooled conveyor).

In the sintering process, abrasive particles can be formed fromrefractory powders having a particle size of up to 10 micrometers indiameter. Binders can be added to the powders along with a lubricant anda suitable solvent, e.g., water. The resulting mixture, mixtures, orslurries can be shaped into platelets or rods of various lengths anddiameters. See, for example, U.S. Pat. No. 3,079,242 (disclosing amethod of making abrasive particles from calcined bauxite materialincluding (1) reducing the material to a fine powder, (2) compactingunder affirmative pressure and forming the fine particles of said powderinto grain sized agglomerations, and (3) sintering the agglomerations ofparticles at a temperature below the fusion temperature of the bauxiteto induce limited recrystallization of the particles, whereby abrasivegrains are produced directly to size).

Chemical ceramic technology involves converting a colloidal dispersionor hydrosol (sometimes called a sol), optionally in a mixture, withsolutions of other metal oxide precursors, into a gel or any otherphysical state that restrains the mobility of the components, drying,and firing to obtain a ceramic material. See, for example, U.S. Pat.Nos. 4,744,802 and 4,848,041. Other relevant disclosures on shapedabrasive particles and associated methods of forming and abrasivearticles incorporating such particles are available at:abel-ip.com/publications/.

Still, there remains a need in the industry for improving performance,life, and efficacy of abrasive particles, and the abrasive articles thatemploy abrasive particles.

SUMMARY

According to one aspect, a method of making a shaped abrasive particlecomprising forming a body of a shaped abrasive comprising at least oneof i) selecting a material having a predetermined strength and formingthe body of the shaped abrasive particle with a predetermined tipsharpness and predetermined Shape Index based upon the predeterminedstrength, ii) selecting a predetermined Shape Index of the body of theshaped abrasive particle and forming the body with the predetermined tipsharpness and the predetermined strength based upon the predeterminedShape Index, and iii) selecting a predetermined tip sharpness of a bodyof the shaped abrasive particle and forming the body of a shapedabrasive particle with a predetermined Shape Index and a predeterminedstrength based upon the predetermined tip sharpness.

In yet another aspect, a shaped abrasive particle includes a bodycomprising a first major surface, a second major surface, and a sidesurface extending between the first major surface and the second majorsurface, wherein the body comprises a sharpness-shape-strength factor(3SF) within a range between about 0.7 and about 1.7 and a Shape Indexwithin a range between at least about 0.51 and not greater than about0.99.

According to another aspect, a shaped abrasive particle includes a bodycomprising a first major surface, a second major surface, and a sidesurface extending between the first major surface and the second majorsurface, wherein the body comprises a Shape Index within a range betweenat least about 0.51 and not greater than about 0.99 and a strength ofnot greater than about 600 MPa and at least about 100 MPa.

Still, for another aspect, a shaped abrasive particle has a bodycomprising a first major surface, a second major surface, and a sidesurface extending between the first major surface and the second majorsurface, wherein the body comprises at least four corners, wherein eachcorner comprises a tip sharpness within a range between not greater thanabout 80 microns and at least about 1 micron, a Shape Index within arange between at least about 0.51 and not greater than about 0.99, andthe body comprises a strength of not greater than about 600 MPa and atleast about 100 MPa.

BRIEF DESCRIPTION OF THE DRAWINGS

The present disclosure may be better understood, and its numerousfeatures and advantages made apparent to those skilled in the art byreferencing the accompanying drawings.

FIG. 1 includes a portion of a system for forming a particulate materialin accordance with an embodiment.

FIG. 2 includes a portion of the system of FIG. 1 for forming aparticulate material in accordance with an embodiment.

FIG. 3 includes a cross-sectional illustration of a shaped abrasiveparticle for illustration of certain features according to embodiments.

FIG. 4 includes a side view of a shaped abrasive particle and percentageflashing according to an embodiment.

FIG. 5A includes an illustration of a bonded abrasive articleincorporating shaped abrasive particles in accordance with anembodiment.

FIG. 5B includes a cross-sectional illustration of a portion of a coatedabrasive article according to an embodiment.

FIG. 6 includes a cross-sectional illustration of a portion of a coatedabrasive article according to an embodiment.

FIG. 7 includes a top-down illustration of a portion of a coatedabrasive article according to an embodiment.

FIG. 8A includes a top-down illustration of a portion of a coatedabrasive article according to an embodiment.

FIG. 8B includes a perspective view illustration of a portion of acoated abrasive article according to an embodiment.

FIG. 9 includes a perspective view illustration of a portion of a coatedabrasive article according to an embodiment.

FIG. 10 includes a top view illustration of a portion of an abrasivearticle in accordance with an embodiment.

FIG. 11 includes images representative of portions of a coated abrasiveaccording to an embodiment and used to analyze the orientation of shapedabrasive particles on the backing.

FIGS. 12A-12C include illustrations of shaped abrasive particles inaccordance with embodiments.

FIGS. 13A-13C include illustrations of shaped abrasive particles inaccordance with embodiments.

FIG. 13D includes a top-down image of a shaped abrasive particle with aline of sectioning for measurement of a draft angle according to anembodiment.

FIG. 13E includes a cross-sectional image of a shaped abrasive particlefor measurement of a draft angle according to an embodiment.

FIG. 13F includes a cross-sectional image of a shaped abrasive particlefor measurement of a draft angle according to an embodiment.

FIGS. 14-26 include illustrations of shaped abrasive particles inaccordance with embodiments.

FIGS. 27-29 include top down images of shaped abrasive particlesaccording to an embodiment.

FIG. 30 includes a top-down image of a conventional shaped abrasiveparticle.

FIG. 31 includes a top-down image of a shaped abrasive particleaccording to an embodiment.

FIG. 32 includes a plot of median force per total area removed from theworkpiece for shaped abrasive particles of Example 1.

FIG. 33 includes a plot of specific grinding energy versus cumulativematerial removed from a workpiece for a representative sample and aconventional sample.

DETAILED DESCRIPTION

The following is directed to abrasive articles including shaped abrasiveparticles. The methods herein may be utilized in forming shaped abrasiveparticles and using abrasive articles incorporating shaped abrasiveparticles. The shaped abrasive particles may be utilized in variousapplications, including for example coated abrasives, bonded abrasives,free abrasives, and a combination thereof. Various other uses may bederived for the shaped abrasive particles.

Shaped Abrasive Particles

Various methods may be utilized to obtain shaped abrasive particles. Theparticles may be obtained from a commercial source or fabricated. Somesuitable processes used to fabricate the shaped abrasive particles caninclude, but is not limited to, depositing, printing (e.g.,screen-printing), molding, pressing, casting, sectioning, cutting,dicing, punching, pressing, drying, curing, coating, extruding, rolling,and a combination thereof. Shaped abrasive particles are formed suchthat each particle has substantially the same arrangement of surfacesand edges relative to each other for shaped abrasive particles havingthe same two-dimensional and three-dimensional shapes. As such, shapedabrasive particles can have a high shape fidelity and consistency in thearrangement of the surfaces and edges relative to other shaped abrasiveparticles of the group having the same two-dimensional andthree-dimensional shape. By contrast, non-shaped abrasive particles canbe formed through different process and have different shape attributes.For example, non-shaped abrasive particles are typically formed by acomminution process, wherein a mass of material is formed and thencrushed and sieved to obtain abrasive particles of a certain size.However, a non-shaped abrasive particle will have a generally randomarrangement of the surfaces and edges, and generally will lack anyrecognizable two-dimensional or three dimensional shape in thearrangement of the surfaces and edges around the body. Moreover,non-shaped abrasive particles of the same group or batch generally lacka consistent shape with respect to each other, such that the surfacesand edges are randomly arranged when compared to each other. Therefore,non-shaped grains or crushed grains have a significantly lower shapefidelity compared to shaped abrasive particles.

FIG. 1 includes an illustration of a system 150 for forming a shapedabrasive particle in accordance with one, non-limiting embodiment. Theprocess of forming shaped abrasive particles can be initiated by forminga mixture 101 including a ceramic material and a liquid. In particular,the mixture 101 can be a gel formed of a ceramic powder material and aliquid. In accordance with an embodiment, the gel can be formed of theceramic powder material as an integrated network of discrete particles.

The mixture 101 may contain a certain content of solid material, liquidmaterial, and additives such that it has suitable rheologicalcharacteristics for use with the process detailed herein. That is, incertain instances, the mixture can have a certain viscosity, and moreparticularly, suitable rheological characteristics that form adimensionally stable phase of material that can be formed through theprocess as noted herein. A dimensionally stable phase of material is amaterial that can be formed to have a particular shape and substantiallymaintain the shape for at least a portion of the processing subsequentto forming. In certain instances, the shape may be retained throughoutsubsequent processing, such that the shape initially provided in theforming process is present in the finally-formed object. It will beappreciated that in some instances, the mixture 101 may not be ashape-stable material, and the process may rely upon solidification andstabilization of the mixture 101 by further processing, such as drying.

The mixture 101 can be formed to have a particular content of solidmaterial, such as the ceramic powder material. For example, in oneembodiment, the mixture 101 can have a solids content of at least about25 wt %, such as at least about 35 wt %, or even at least about 38 wt %for the total weight of the mixture 101. Still, in at least onenon-limiting embodiment, the solids content of the mixture 101 can benot greater than about 75 wt %, such as not greater than about 70 wt %,not greater than about 65 wt %, not greater than about 55 wt %, notgreater than about 45 wt %, or not greater than about 42 wt %. It willbe appreciated that the content of the solids materials in the mixture101 can be within a range between any of the minimum and maximumpercentages noted above.

According to one embodiment, the ceramic powder material can include anoxide, a nitride, a carbide, a boride, an oxycarbide, an oxynitride, anda combination thereof. In particular instances, the ceramic material caninclude alumina. More specifically, the ceramic material may include aboehmite material, which may be a precursor of alpha alumina. The term“boehmite” is generally used herein to denote alumina hydrates includingmineral boehmite, typically being Al₂O₃.H₂O and having a water contenton the order of 15%, as well as pseudoboehmite, having a water contenthigher than 15%, such as 20-38% by weight. It is noted that boehmite(including pseudoboehmite) has a particular and identifiable crystalstructure, and therefore a unique X-ray diffraction pattern. As such,boehmite is distinguished from other aluminous materials including otherhydrated aluminas such as ATH (aluminum trihydroxide), a commonprecursor material used herein for the fabrication of boehmiteparticulate materials.

Furthermore, the mixture 101 can be formed to have a particular contentof liquid material. Some suitable liquids may include water. Inaccordance with one embodiment, the mixture 101 can be formed to have aliquid content less than the solids content of the mixture 101. In moreparticular instances, the mixture 101 can have a liquid content of atleast about 25 wt % for the total weight of the mixture 101. In otherinstances, the amount of liquid within the mixture 101 can be greater,such as at least about 35 wt %, at least about 45 wt %, at least about50 wt %, or even at least about 58 wt %. Still, in at least onenon-limiting embodiment, the liquid content of the mixture can be notgreater than about 75 wt %, such as not greater than about 70 wt %, notgreater than about 65 wt %, not greater than about 62 wt %, or even notgreater than about 60 wt %. It will be appreciated that the content ofthe liquid in the mixture 101 can be within a range between any of theminimum and maximum percentages noted above.

Furthermore, to facilitate processing and forming shaped abrasiveparticles according to embodiments herein, the mixture 101 can have aparticular storage modulus. For example, the mixture 101 can have astorage modulus of at least about 1×10⁴ Pa, such as at least about 4×10⁴Pa, or even at least about 5×10⁴ Pa. However, in at least onenon-limiting embodiment, the mixture 101 may have a storage modulus ofnot greater than about 1×10⁷ Pa, such as not greater than about 2×10⁶Pa. It will be appreciated that the storage modulus of the mixture 101can be within a range between any of the minimum and maximum valuesnoted above.

The storage modulus can be measured via a parallel plate system usingARES or AR-G2 rotational rheometers, with Peltier plate temperaturecontrol systems. For testing, the mixture 101 can be extruded within agap between two plates that are set to be approximately 8 mm apart fromeach other. After extruding the gel into the gap, the distance betweenthe two plates defining the gap is reduced to 2 mm until the mixture 101completely fills the gap between the plates. After wiping away excessmixture, the gap is decreased by 0.1 mm and the test is initiated. Thetest is an oscillation strain sweep test conducted with instrumentsettings of a strain range between 0.01% to 100%, at 6.28 rad/s (1 Hz),using 25-mm parallel plate and recording 10 points per decade. Within 1hour after the test completes, the gap is lowered again by 0.1 mm andthe test is repeated. The test can be repeated at least 6 times. Thefirst test may differ from the second and third tests. Only the resultsfrom the second and third tests for each specimen should be reported.

Furthermore, to facilitate processing and forming shaped abrasiveparticles according to embodiments herein, the mixture 101 can have aparticular viscosity. For example, the mixture 101 can have a viscosityof at least about 2×10³ Pa s, such as at least about 3×10³ Pa s, atleast about 4×10³ Pa s, at least about 5×10³ Pa s, at least about 6×10³Pa s, at least about 8×10³ Pa s, at least about 10×10³ Pa s, at leastabout 20×10³ Pa s, at least about 30×10³ Pa s, at least about 40×10³ Pas, at least about 50×10³ Pa s, at least about 60×10³ Pa s, or at leastabout 65×10³ Pa s. In at least one non-limiting embodiment, the mixture101 may have a viscosity of not greater than about 100×10³ Pa s, such asnot greater than about 95×10³ Pa s, not greater than about 90×10³ Pa s,or even not greater than about 85×10³ Pa s. It will be appreciated thatthe viscosity of the mixture 101 can be within a range between any ofthe minimum and maximum values noted above. The viscosity can bemeasured in the same manner as the storage modulus as described above.

Moreover, the mixture 101 can be formed to have a particular content oforganic materials including, for example, organic additives that can bedistinct from the liquid to facilitate processing and formation ofshaped abrasive particles according to the embodiments herein. Somesuitable organic additives can include stabilizers, binders such asfructose, sucrose, lactose, glucose, UV curable resins, and the like.

Notably, the embodiments herein may utilize a mixture 101 that can bedistinct from slurries used in conventional forming operations. Forexample, the content of organic materials within the mixture 101 and, inparticular, any of the organic additives noted above, may be a minoramount as compared to other components within the mixture 101. In atleast one embodiment, the mixture 101 can be formed to have not greaterthan about 30 wt % organic material for the total weight of the mixture101. In other instances, the amount of organic materials may be less,such as not greater than about 15 wt %, not greater than about 10 wt %,or even not greater than about 5 wt %. Still, in at least onenon-limiting embodiment, the amount of organic materials within themixture 101 can be at least about 0.01 wt %, such as at least about 0.5wt % for the total weight of the mixture 101. It will be appreciatedthat the amount of organic materials in the mixture 101 can be within arange between any of the minimum and maximum values noted above.

Moreover, the mixture 101 can be formed to have a particular content ofacid or base, distinct from the liquid content, to facilitate processingand formation of shaped abrasive particles according to the embodimentsherein. Some suitable acids or bases can include nitric acid, sulfuricacid, citric acid, chloric acid, tartaric acid, phosphoric acid,ammonium nitrate, and ammonium citrate. According to one particularembodiment in which a nitric acid additive is used, the mixture 101 canhave a pH of less than about 5, and more particularly, can have a pHwithin a range between about 2 and about 4.

The system 150 of FIG. 1, can include a die 103. As illustrated, themixture 101 can be provided within the interior of the die 103 andconfigured to be extruded through a die opening 105 positioned at oneend of the die 103. As further illustrated, extruding can includeapplying a force 180 on the mixture 101 to facilitate extruding themixture 101 through the die opening 105. During extrusion within anapplication zone 183, a tool 151 can be in direct contact with a portionof the die 103 and facilitate extrusion of the mixture 101 into the toolcavities 152. The tool 151 can be in the form of a screen, such asillustrated in FIG. 1, wherein the cavities 152 extend through theentire thickness of the tool 151. Still, it will be appreciated that thetool 151 may be formed such that the cavities 152 extend for a portionof the entire thickness of the tool 151 and have a bottom surface, suchthat the volume of space configured to hold and shape the mixture 101 isdefined by a bottom surface and side surfaces.

The tool 151 may be formed of a metal material, including for example, ametal alloy, such as stainless steel. In other instances, the tool 151may be formed of an organic material, such as a polymer.

In accordance with an embodiment, a particular pressure may be utilizedduring extrusion. For example, the pressure can be at least about 10kPa, such as at least about 500 kPa. Still, in at least one non-limitingembodiment, the pressure utilized during extrusion can be not greaterthan about 4 MPa. It will be appreciated that the pressure used toextrude the mixture 101 can be within a range between any of the minimumand maximum values noted above. In particular instances, the consistencyof the pressure delivered by a piston 199 may facilitate improvedprocessing and formation of shaped abrasive particles. Notably,controlled delivery of consistent pressure across the mixture 101 andacross the width of the die 103 can facilitate improved processingcontrol and improved dimensional characteristics of the shaped abrasiveparticles.

Prior to depositing the mixture 101 in the tool cavities 152, a moldrelease agent can be applied to the surfaces of the tool cavities 152,which may facilitate removal of precursor shaped abrasive particles fromthe tool cavities 152 after further processing. Such a process can beoptional and may not necessarily be used to conduct the molding process.A suitable exemplary mold release agent can include an organic material,such as one or more polymers (e.g., PTFE). In other instances, an oil(synthetic or organic) may be applied as a mold release agent to thesurfaces of the tool cavities 152. One suitable oil may be peanut oil.The mold release agent may be applied using any suitable manner,including but not limited to, depositing, spraying, printing, brushing,coating, and the like.

The mixture 101 may be deposited within the tool cavities 152, which maybe shaped in any suitable manner to form shaped abrasive particleshaving shapes corresponding to the shape of the tool cavities 152.

Referring briefly to FIG. 2, a portion of the tool 151 is illustrated.As shown, the tool 151 can include the tool cavities 152, and moreparticularly, a plurality of tool cavities 152 extending into the volumeof the tool 151. In accordance with an embodiment, the tool cavities 152can have a two-dimensional shape as viewed in a plane defined by thelength (l) and width (w) of the tool 151. The two-dimensional shape caninclude various shapes such as, for example, polygons, ellipsoids,numerals, Greek alphabet letters, Latin alphabet letters, Russianalphabet characters, complex shapes including a combination of polygonalshapes, and a combination thereof. In particular instances, the toolcavities 152 may have two-dimensional polygonal shapes such as arectangle, a quadrilateral, a pentagon, a hexagon, a heptagon, anoctagon, a nonagon, a decagon, and a combination thereof. Notably, aswill be appreciated in further reference to the shaped abrasiveparticles of the embodiments herein, the tool cavities 152 may utilizevarious other shapes.

While the tool 151 of FIG. 2 is illustrated as having tool cavities 152oriented in a particular manner relative to each other, it will beappreciated that various other orientations may be utilized. Inaccordance with one embodiment, each of the tool cavities 152 can havesubstantially the same orientation relative to each other, andsubstantially the same orientation relative to the surface of thescreen. For example, each of the tool cavities 152 can have a first edge154 defining a first plane 155 for a first row 156 of the tool cavities152 extending laterally across a lateral axis 158 of the tool 151. Thefirst plane 155 can extend in a direction substantially orthogonal to alongitudinal axis 157 of the tool 151. However, it will be appreciated,that in other instances, the tool cavities 152 need not necessarily havethe same orientation relative to each other.

Moreover, the first row 156 of tool cavities 152 can be orientedrelative to a direction of translation to facilitate particularprocessing and controlled formation of shaped abrasive particles. Forexample, the tool cavities 152 can be arranged on the tool 151 such thatthe first plane 155 of the first row 156 defines an angle relative tothe direction of translation 171. As illustrated, the first plane 155can define an angle that is substantially orthogonal to the direction oftranslation 171. Still, it will be appreciated that in one embodiment,the tool cavities 152 can be arranged on the tool 151 such that thefirst plane 155 of the first row 156 defines a different angle withrespect to the direction of translation, including for example, an acuteangle or an obtuse angle. Still, it will be appreciated that the toolcavities 152 may not necessarily be arranged in rows. The tool cavities152 may be arranged in various particular ordered distributions withrespect to each other on the tool 151, such as in the form of atwo-dimensional pattern. Alternatively, the openings may be disposed ina random manner on the tool 151.

Referring again to FIG. 1, during operation of the system 150, the tool151 can be translated in a direction 153 to facilitate a continuousmolding operation. As will be appreciated, the tool 151 may be in theform of a continuous belt, which can be translated over rollers tofacilitate continuous processing. In some embodiments, the tool 151 canbe translated while extruding the mixture 101 through the die opening105. As illustrated in the system 150, the mixture 101 may be extrudedin a direction 191. The direction of translation 153 of the tool 151 canbe angled relative to the direction of extrusion 191 of the mixture 101.While the angle between the direction of translation 153 and thedirection of extrusion 191 is illustrated as substantially orthogonal inthe system 100, other angles are contemplated, including for example, anacute angle or an obtuse angle. After the mixture 101 is extrudedthrough the die opening 105, the mixture 101 and tool 151 may betranslated under a knife edge 107 attached to a surface of the die 103.The knife edge 107 may define a region at the front of the die 103 thatfacilitates displacement of the mixture 101 into the tool cavities 152of the tool 151.

In the molding process, the mixture 101 may undergo significant dryingwhile contained in the tool cavity 152. Therefore, shaping may beprimarily attributed to substantial drying and solidification of themixture 101 in the tool cavities 152 to shape the mixture 101. Incertain instances, the shaped abrasive particles formed according to themolding process may exhibit shapes more closely replicating the featuresof the mold cavity compared to other processes, including for example,screen printing processes. However, it should be noted that certainbeneficial shape characteristics may be more readily achieved throughscreen printing processes.

After applying the mold release agent, the mixture 101 can be depositedwithin the mold cavities and dried. Drying may include removal of aparticular content of certain materials from the mixture 101, includingvolatiles, such as water or organic materials. In accordance with anembodiment, the drying process can be conducted at a drying temperatureof not greater than about 300° C., such as not greater than about 250°C., not greater than about 200° C., not greater than about 150° C., notgreater than about 100° C., not greater than about 80° C., not greaterthan about 60° C., not greater than about 40° C., or even not greaterthan about 30° C. Still, in one non-limiting embodiment, the dryingprocess may be conducted at a drying temperature of at least about −20°C., such as at least about −10° C. at least about 0° C. at least about5° C. at least about 10° C., or even at least about 20° C. It will beappreciated that the drying temperature may be within a range betweenany of the minimum and maximum temperatures noted above.

In certain instances, drying may be conducted for a particular durationto facilitate the formation of shaped abrasive particles according toembodiments herein. For example, drying can be conducted for a durationof at least about 1 minute, such as at least about 2 minutes, at leastabout 4 minutes, at least about 6 minutes, at least about 8 minutes, atleast about 10 minutes, such as at least about 30 minutes, at leastabout 1 hour, at least about 2 hours, at least about 4 hours, at leastabout 8 hours, at least about 12 hours, at least about 15 hours, atleast about 18 hours, at least about 24 hours. In still other instances,the process of drying may be not greater than about 30 hours, such asnot greater than about 24 hours, not greater than about 20 hours, notgreater than about 15 hours, not greater than about 12 hours, notgreater than about 10 hours, not greater than about 8 hours, not greaterthan about 6 hours, not greater than about 4 hours. It will beappreciated that the duration of drying can be within a range betweenany of the minimum and maximum values noted above.

Additionally, drying may be conducted at a particular relative humidityto facilitate formation of shaped abrasive particles according to theembodiments herein. For example, drying may be conducted at a relativehumidity of at least about 20%, at least about 30%, at least about 40%,at least about 50%, at least about 60%, such as at least about 62%, atleast about 64%, at least about 66%, at least about 68%, at least about70%, at least about 72%, at least about 74%, at least about 76%, atleast about 78%, or even at least about 80%. In still other non-limitingembodiments, drying may be conducted at a relative humidity of notgreater than about 90%, such as not greater than about 88%, not greaterthan about 86%, not greater than about 84%, not greater than about 82%,not greater than about 80%, not greater than about 78%, not greater thanabout 76%, not greater than about 74%, not greater than about 72%, notgreater than about 70%, not greater than about 65%, not greater thanabout 60%, not greater than about 55%, not greater than about 50%, notgreater than about 45%, not greater than about 40%, not greater thanabout 35%, not greater than about 30%, or even not greater than about25%. It will be appreciated that the relative humidity utilized duringdrying can be within a range between any of the minimum and maximumpercentages noted above.

After completing the drying process, the mixture 101 can be releasedfrom the tool cavities 152 to produce precursor shaped abrasiveparticles. Notably, before the mixture 101 is removed from the toolcavities 152 or after the mixture 101 is removed and the precursorshaped abrasive particles are formed, one or more post-forming processesmay be completed. Such processes can include surface shaping, curing,reacting, radiating, planarizing, calcining, sintering, sieving, doping,and a combination thereof. For example, in one optional process, themixture 101 or precursor shaped abrasive particles may be translatedthrough an optional shaping zone, wherein at least one exterior surfaceof the mixture or precursor shaped abrasive particles may be shaped. Instill another embodiment, the mixture 101 as contained in the moldcavities or the precursor shaped abrasive particles may be translatedthrough an optional application zone, wherein a dopant material can beapplied. In particular instances, the process of applying a dopantmaterial can include selective placement of the dopant material on atleast one exterior surface of the mixture 101 or precursor shapedabrasive particles.

The dopant material may be applied utilizing various methods includingfor example, spraying, dipping, depositing, impregnating, transferring,punching, cutting, pressing, crushing, and any combination thereof. Inaccordance with an embodiment, applying a dopant material can includethe application of a particular material, such as a precursor. Incertain instances, the precursor can be a salt, such as a metal salt,that includes a dopant material to be incorporated into thefinally-formed shaped abrasive particles. For example, the metal saltcan include an element or compound that is the precursor to the dopantmaterial. It will be appreciated that the salt material may be in liquidform, such as in a dispersion comprising the salt and liquid carrier.The salt may include nitrogen, and more particularly, can include anitrate. In other embodiments, the salt can be a chloride, sulfate,phosphate, and a combination thereof. In one embodiment, the salt caninclude a metal nitrate, and more particularly, consist essentially of ametal nitrate. In one embodiment, the dopant material can include anelement or compound such as an alkali element, alkaline earth element,rare earth element, hafnium, zirconium, niobium, tantalum, molybdenum,vanadium, or a combination thereof. In one particular embodiment, thedopant material includes an element or compound including an elementsuch as lithium, sodium, potassium, magnesium, calcium, strontium,barium, scandium, yttrium, lanthanum, cesium, praseodymium, niobium,hafnium, zirconium, tantalum, molybdenum, vanadium, chromium, cobalt,iron, germanium, manganese, nickel, titanium, zinc, and a combinationthereof.

The molding process may further include a sintering process. For certainembodiments herein, sintering can be conducted after removing themixture from the tool cavities 152 and forming the precursor shapedabrasive particles. Sintering of the precursor shaped abrasive particles123 may be utilized to densify the particles, which are generally in agreen state. In a particular instance, the sintering process canfacilitate the formation of a high-temperature phase of the ceramicmaterial. For example, in one embodiment, the precursor shaped abrasiveparticles may be sintered such that a high-temperature phase of alumina,such as alpha alumina, is formed. In one instance, a shaped abrasiveparticle can comprise at least about 90 wt % alpha alumina for the totalweight of the particle. In other instances, the content of alpha aluminamay be greater such that the shaped abrasive particle may consistessentially of alpha alumina.

The body of the finally-formed shaped abrasive particles can haveparticular two-dimensional shapes. For example, the body can have atwo-dimensional shape, as viewed in a plane defined by the length andwidth of the body, and can have a shape including a polygonal shape,ellipsoidal shape, a numeral, a Greek alphabet character, a Latinalphabet character, a Russian alphabet character, a complex shapeutilizing a combination of polygonal shapes and a combination thereof.Particular polygonal shapes include rectangular, trapezoidal,pentagonal, hexagonal, heptagonal, octagonal, nonagonal, decagonal, andany combination thereof. In another instance, the finally-formed shapedabrasive particles can have a body having a two-dimensional shape suchas an irregular quadrilateral, an irregular rectangle, an irregulartrapezoid, an irregular pentagon, an irregular hexagon, an irregularheptagon, an irregular octagon, an irregular nonagon, an irregulardecagon, and a combination thereof. An irregular polygonal shape is onewhere at least one of the sides defining the polygonal shape isdifferent in dimension (e.g., length) with respect to another side. Asillustrated in other embodiments herein, the two-dimensional shape ofcertain shaped abrasive particles can have a particular number ofexterior points or external corners. For example, the body of the shapedabrasive particles can have a two-dimensional polygonal shape as viewedin a plane defined by a length and width, wherein the body comprises atwo-dimensional shape having at least 4 exterior points (e.g., aquadrilateral), at least 5 exterior points (e.g., a pentagon), at least6 exterior points (e.g., a hexagon), at least 7 exterior points (e.g., aheptagon), at least 8 exterior points (e.g., an octagon), at least 9exterior points (e.g., a nonagon), and the like.

FIG. 3 includes a cross-sectional illustration of a shaped abrasiveparticle to illustrate certain features of shaped abrasive particles ofthe embodiments herein. It will be appreciated that such across-sectional view can be applied to any of the exemplary shapedabrasive particles of the embodiments to determine one or more shapeaspects or dimensional characteristics as described herein. The body ofthe shaped abrasive particle can include an upper major surface 303(i.e., a first major surface) and a bottom major surface 304 (i.e., asecond major surface) opposite the upper major surface 303. The uppersurface 303 and the bottom surface 304 can be separated from each otherby a side surface 314.

In certain instances, the shaped abrasive particles of the embodimentsherein can have an average difference in height, which is a measure ofthe difference between hc and hm. Notably, the dimension of Lmiddle canbe a length defining a distance between a height at a corner (hc) and aheight at a midpoint edge (hm) opposite the corner. Moreover, the body301 can have an interior height (hi), which can be the smallestdimension of height of the body 301 as measured along a dimensionbetween any corner and opposite midpoint edge on the body 301. Forconvention herein, average difference in height will be generallyidentified as hc-hm, however it is defined as an absolute value of thedifference. Therefore, it will be appreciated that average difference inheight may be calculated as hm-hc when the height of the body 301 at theside surface 314 is greater than the height at the corner 313. Moreparticularly, the average difference in height can be calculated basedupon a plurality of shaped abrasive particles from a suitable samplesize. The heights hc and hm of the particles can be measured using aSTIL (Sciences et Techniques Industrielles de la Lumiere—France) MicroMeasure 3D Surface Profilometer (white light (LED) chromatic aberrationtechnique) and the average difference in height can be calculated basedon the average values of hc and hm from the sample.

As illustrated in FIG. 3, in one particular embodiment, the body 301 ofthe shaped abrasive particle 300 may have an average difference inheight at different locations at the body 301. The body 301 can have anaverage difference in height, which can be the absolute value of [hc−hm]between the first corner height (hc) and the second midpoint height (hm)that is quite low, such that the particle is relative flat, having anaverage difference in height that is not greater than about 300 microns,such as not greater than about 250 microns, not greater than about 220microns, not greater than about 180 microns, not greater than about 150microns, not greater than about 100 microns, not greater than about 50microns, or even not greater than about 20 microns.

The body of the shaped abrasive particles herein can include a width (w)that is the longest dimension of the body and extending along a side.The shaped abrasive particles can include a length that extends througha midpoint (which may be along a major surface) of the body andbisecting the body (i.e., Lmiddle). The body can further include aheight (h), which may be a dimension of the body extending in adirection perpendicular to the length and width in a direction definedby a side surface of the body 301. In specific instances, the width canbe greater than or equal to the length, the length can be greater thanor equal to the height, and the width can be greater than or equal tothe height.

In particular instances, the body 301 can be formed to have a primaryaspect ratio, which is a ratio expressed as width:length, having a valueof at least 1:1. In other instances, the body 301 can be formed suchthat the primary aspect ratio (w:l) is at least about 1.5:1, such as atleast about 2:1, at least about 4:1, or even at least about 5:1. Still,in other instances, the abrasive particle 300 can be formed such thatthe body 301 has a primary aspect ratio that is not greater than about10:1, such as not greater than 9:1, not greater than about 8:1, or evennot greater than about 5:1. It will be appreciated that the body 301 canhave a primary aspect ratio within a range between any of the ratiosnoted above. Furthermore, it will be appreciated that reference hereinto a height can be reference to the maximum height measurable of theabrasive particle 300.

In addition to the primary aspect ratio, the abrasive particle 300 canbe formed such that the body 301 comprises a secondary aspect ratio,which can be defined as a ratio of length:height, wherein the height isan interior median height (Mhi). In certain instances, the secondaryaspect ratio can be at least about 1:1, such as at least about 2:1, atleast about 4:1, or even at least about 5:1. Still, in other instances,the abrasive particle 300 can be formed such that the body 301 has asecondary aspect ratio that is not greater than about 1:3, such as notgreater than 1:2, or even not greater than about 1:1. It will beappreciated that the body 301 can have a secondary aspect ratio within arange between any of the ratios noted above, such as within a rangebetween about 5:1 and about 1:1.

In accordance with another embodiment, the abrasive particle 300 can beformed such that the body 301 comprises a tertiary aspect ratio, definedby the ratio width:height, wherein the height is an interior medianheight (Mhi). The tertiary aspect ratio of the body 301 can be can be atleast about 1:1, such as at least about 2:1, at least about 4:1, atleast about 5:1, or even at least about 6:1. Still, in other instances,the abrasive particle 300 can be formed such that the body 301 has atertiary aspect ratio that is not greater than about 3:1, such as notgreater than 2:1, or even not greater than about 1:1. It will beappreciated that the body 301 can have a tertiary aspect ratio within arange between any of the ratios noted above, such as within a rangebetween about 6:1 and about 1:1.

According to one embodiment, the body 301 of the shaped abrasiveparticle 300 can have particular dimensions, which may facilitateimproved performance. For example, in one instance, the body 301 canhave an interior height (hi), which can be the smallest dimension ofheight of the body 301 as measured along a dimension between any cornerand opposite midpoint edge on the body 301. In particular instances, theinterior height (hi) may be the smallest dimension of height (i.e.,measure between the bottom surface 304 and the upper surface 305) of thebody 301 for three measurements taken between each of the exteriorcorners and the opposite midpoint edges. The interior height (hi) of thebody 301 of a shaped abrasive particle 300 is illustrated in FIG. 3.According to one embodiment, the interior height (hi) can be at leastabout 20% of the width (w). The height (hi) may be measured bysectioning or mounting and grinding the shaped abrasive particle 300 andviewing in a manner sufficient (e.g., light microscope or SEM) todetermine the smallest height (hi) within the interior of the body 301.In one particular embodiment, the height (hi) can be at least about 22%of the width, such as at least about 25%, at least about 30%, or even atleast about 33%, of the width of the body 301. For one non-limitingembodiment, the height (hi) of the body 301 can be not greater thanabout 80% of the width of the body 301, such as not greater than about76%, not greater than about 73%, not greater than about 70%, not greaterthan about 68% of the width, not greater than about 56% of the width,not greater than about 48% of the width, or even not greater than about40% of the width. It will be appreciated that the height (hi) of thebody 301 can be within a range between any of the above noted minimumand maximum percentages.

A batch of shaped abrasive particles can be fabricated where the medianinterior height value (Mhi) can be controlled, which may facilitateimproved performance. In particular, the median internal height (hi) ofa batch can be related to a median width of the shaped abrasiveparticles of the batch in the same manner as described above. Notably,the median interior height (Mhi) can be at least about 20% of the width,such as at least about 22%, at least about 25%, at least about 30%, oreven at least about 33% of the median width of the shaped abrasiveparticles of the batch. For one non-limiting embodiment, the medianinterior height (Mhi) of the body 301 can be not greater than about 80%,such as not greater than about 76%, not greater than about 73%, notgreater than about 70%, not greater than about 68% of the width, notgreater than about 56% of the width, not greater than about 48% of thewidth, or even not greater than about 40% of the median width of thebody 301. It will be appreciated that the median interior height (Mhi)of the body 301 can be within a range between any of the above notedminimum and maximum percentages.

Furthermore, the batch of shaped abrasive particles may exhibit improveddimensional uniformity as measured by the standard deviation of adimensional characteristic from a suitable sample size. According to oneembodiment, the shaped abrasive particles can have an interior heightvariation (Vhi), which can be calculated as the standard deviation ofinterior height (hi) for a suitable sample size of particles from abatch. According to one embodiment, the interior height variation can benot greater than about 60 microns, such as not greater than about 58microns, not greater than about 56 microns, or even not greater thanabout 54 microns. In one non-limiting embodiment, the interior heightvariation (Vhi) can be at least about 2 microns. It will be appreciatedthat the interior height variation of the body can be within a rangebetween any of the above noted minimum and maximum values.

For another embodiment, the body 301 of the shaped abrasive particle 300can have a height, which may be an interior height (hi), of at leastabout 70 microns. More particularly, the height may be at least about 80microns, such as at least about 90 microns, at least about 100 microns,at least about 110 microns, at least about 120 microns, at least about150 microns, at least about 175 microns, at least about 200 microns, atleast about 225 microns, at least about 250 microns, at least about 275microns, or even at least about 300 microns. In still one non-limitingembodiment, the height of the body 301 can be not greater than about 3mm, such as not greater than about 2 mm, not greater than about 1.5 mm,not greater than about 1 mm, or even not greater than about 800 microns,not greater than about 600 microns, not greater than about 500 microns,not greater than about 475 microns, not greater than about 450 microns,not greater than about 425 microns, not greater than about 400 microns,not greater than about 375 microns, not greater than about 350 microns,not greater than about 325 microns, not greater than about 300 microns,not greater than about 275 microns, or even not greater than about 250microns. It will be appreciated that the height of the body 301 can bewithin a range between any of the above noted minimum and maximumvalues. Moreover, it will be appreciated that the above range of valuescan be representative of a median interior height (Mhi) value for abatch of shaped abrasive particles.

For certain embodiments herein, the body 301 of the shaped abrasiveparticle 300 can have particular dimensions, including for example, awidth≧length, a length≧height, and a width≧height. More particularly,the body 301 of the shaped abrasive particle 300 can have a width (w) ofat least about 200 microns, such as at least about 250 microns, at leastabout 300 microns, at least about 350 microns, at least about 400microns, at least about 450 microns, at least about 500 microns, atleast about 550 microns, at least about 600 microns, at least about 700microns, at least about 800 microns, or even at least about 900 microns.In one non-limiting instance, the body 301 can have a width of notgreater than about 4 mm, such as not greater than about 3 mm, notgreater than about 2.5 mm, or even not greater than about 2 mm. It willbe appreciated that the width of the body 301 can be within a rangebetween any of the above noted minimum and maximum values. Moreover, itwill be appreciated that the above range of values can be representativeof a median width (Mw) for a batch of shaped abrasive particles.

The body 301 of the shaped abrasive particle 300 can have particulardimensions, including for example, a length (Lmiddle or Lp) of at leastabout 0.4 mm, such as at least about 0.6 mm, at least about 0.8 mm, oreven at least about 0.9 mm. Still, for at least one non-limitingembodiment, the body 301 can have a length of not greater than about 4mm, such as not greater than about 3 mm, not greater than about 2.5 mm,or even not greater than about 2 mm. It will be appreciated that thelength of the body 301 can be within a range between any of the abovenoted minimum and maximum values. Moreover, it will be appreciated thatthe above range of values can be representative of a median length (Ml),which may be more particularly a median middle length (MLmiddle) ormedian profile length (MLp), for a batch of shaped abrasive particles.

The shaped abrasive particle 300 can have a body 301 having a particularamount of dishing, wherein the dishing value (d) can be defined as aratio between an average height of the body 301 at the exterior corners(Ahc) as compared to the smallest dimension of height of the body 301 atthe interior (hi). The average height of the body 301 at the corners(Ahc) can be calculated by measuring the height of the body 301 at allcorners and averaging the values, and may be distinct from a singlevalue of height at one corner (hc). The average height of the body 301at the corners or at the interior can be measured using a STIL (Scienceset Techniques Industrielles de la Lumiere—France) Micro Measure 3DSurface Profilometer (white light (LED) chromatic aberration technique).Alternatively, the dishing may be based upon a median height of theparticles at the corner (Mhc) calculated from a suitable sampling ofparticles from a batch. Likewise, the interior height (hi) can be amedian interior height (Mhi) derived from a suitable sampling of shapedabrasive particles from a batch. According to one embodiment, thedishing value (d) can be not greater than about 2, such as not greaterthan about 1.9, not greater than about 1.8, not greater than about 1.7,not greater than about 1.6, not greater than about 1.5, or even notgreater than about 1.2. Still, in at least one non-limiting embodiment,the dishing value (d) can be at least about 0.9, such as at least about1.0. It will be appreciated that the dishing ratio can be within a rangebetween any of the minimum and maximum values noted above. Moreover, itwill be appreciated that the above dishing values can be representativeof a median dishing value (Md) for a batch of shaped abrasive particles.

The shaped abrasive particles of the embodiments herein, including forexample, the body 301 of the particle of FIG. 3 can have a bottomsurface 304 defining a bottom area (A_(b)). In particular instances, thebottom surface 304 can be the largest surface of the body 301. Thebottom major surface 304 can have a surface area defined as the bottomarea (A_(b)) that is different than the surface area of the upper majorsurface 303. In one particular embodiment, the bottom major surface 304can have a surface area defined as the bottom area (A_(b)) that isdifferent than the surface area of the upper major surface 303. Inanother embodiment, the bottom major surface 304 can have a surface areadefined as the bottom area (A_(b)) that is less than the surface area ofthe upper major surface 303.

Additionally, the body 301 can have a cross-sectional midpoint area(A_(m)) defining an area of a plane perpendicular to the bottom area(A_(b)) and extending through a midpoint 381 of the particle 300. Incertain instances, the body 301 can have an area ratio of bottom area tomidpoint area (A_(b)/A_(m)) of not greater than about 6. In moreparticular instances, the area ratio can be not greater than about 5.5,such as not greater than about 5, not greater than about 4.5, notgreater than about 4, not greater than about 3.5, or even not greaterthan about 3. Still, in one non-limiting embodiment, the area ratio maybe at least about 1.1, such as at least about 1.3, or even at leastabout 1.8. It will be appreciated that the area ratio can be within arange between any of the minimum and maximum values noted above.Moreover, it will be appreciated that the above area ratios can berepresentative of a median area ratio for a batch of shaped abrasiveparticles.

Furthermore the shaped abrasive particles of the embodiments hereinincluding, for example, the particle of FIG. 3, can have a normalizedheight difference of not greater than about 0.3. The normalized heightdifference can be defined by the absolute value of the equation[(hc−hm)/(hi)]. In other embodiments, the normalized height differencecan be not greater than about 0.26, such as not greater than about 0.22,or even not greater than about 0.19. Still, in one particularembodiment, the normalized height difference can be at least about 0.04,such as at least about 0.05, or even at least about 0.06. It will beappreciated that the normalized height difference can be within a rangebetween any of the minimum and maximum values noted above. Moreover, itwill be appreciated that the above normalized height values can berepresentative of a median normalized height value for a batch of shapedabrasive particles.

The shaped abrasive particle 300 can be formed such that the body 301includes a crystalline material, and more particularly, apolycrystalline material. Notably, the polycrystalline material caninclude abrasive grains. In one embodiment, the body 301 can beessentially free of an organic material, including for example, abinder. More particularly, the body 301 can consist essentially of apolycrystalline material.

In one aspect, the body 301 of the shaped abrasive particle 300 can bean agglomerate including a plurality of abrasive particles, grit, and/orgrains bonded to each other to form the body 301 of the abrasiveparticle 300. Suitable abrasive grains can include nitrides, oxides,carbides, borides, oxynitrides, oxyborides, diamond, and a combinationthereof. In particular instances, the abrasive grains can include anoxide compound or complex, such as aluminum oxide, zirconium oxide,titanium oxide, yttrium oxide, chromium oxide, strontium oxide, siliconoxide, and a combination thereof. In one particular instance, theabrasive particle 300 is formed such that the abrasive grains formingthe body 301 include alumina, and more particularly, may consistessentially of alumina. Moreover, in particular instances, the shapedabrasive particle 300 can be formed from a seeded sol-gel.

The abrasive grains (i.e., crystallites) contained within the body 301may have an average grain size that is generally not greater than about100 microns. In other embodiments, the average grain size can be less,such as not greater than about 80 microns, not greater than about 50microns, not greater than about 30 microns, not greater than about 20microns, not greater than about 10 microns, or even not greater thanabout 1 micron, not greater than about 0.9 microns, not greater thanabout 0.8 microns, not greater than about 0.7 microns, or even notgreater than about 0.6 microns. Still, the average grain size of theabrasive grains contained within the body 301 can be at least about 0.01microns, such as at least about 0.05 microns, at least about 0.06microns, at least about 0.07 microns, at least about 0.08 microns, atleast about 0.09 microns, at least about 0.1 microns, at least about0.12 microns, at least about 0.15 microns, at least about 0.17 microns,at least about 0.2 microns, or even at least about 0.5 microns. It willbe appreciated that the abrasive grains can have an average grain sizewithin a range between any of the minimum and maximum values notedabove.

In accordance with certain embodiments, the abrasive particle 300 can bea composite article including at least two different types of grainswithin the body 301. It will be appreciated that different types ofgrains are grains having different compositions with regard to eachother. For example, the body 301 can be formed such that is includes atleast two different types of grains, wherein the two different types ofgrains can be nitrides, oxides, carbides, borides, oxynitrides,oxyborides, diamond, and a combination thereof.

In accordance with an embodiment, the abrasive particle 300 can have anaverage particle size, as measured by the largest dimension measurableon the body 301, of at least about 100 microns. In fact, the abrasiveparticle 300 can have an average particle size of at least about 150microns, such as at least about 200 microns, at least about 300 microns,at least about 400 microns, at least about 500 microns, at least about600 microns, at least about 700 microns, at least about 800 microns, oreven at least about 900 microns. Still, the abrasive particle 300 canhave an average particle size that is not greater than about 5 mm, suchas not greater than about 3 mm, not greater than about 2 mm, or even notgreater than about 1.5 mm. It will be appreciated that the abrasiveparticle 300 can have an average particle size within a range betweenany of the minimum and maximum values noted above.

The shaped abrasive particles of the embodiments herein can have apercent flashing that may facilitate improved performance. Notably, theflashing defines an area of the particle as viewed along one side, suchas illustrated in FIG. 4, wherein the flashing extends from a sidesurface of the body 301 within the boxes 402 and 403. The flashing canrepresent tapered regions proximate to the upper surface 303 and bottomsurface 304 of the body 301. The flashing can be measured as thepercentage of area of the body 301 along the side surface containedwithin a box extending between an innermost point of the side surface(e.g., 421) and an outermost point (e.g., 422) on the side surface ofthe body 301. In one particular instance, the body 301 can have aparticular content of flashing, which can be the percentage of area ofthe body 301 contained within the boxes 402 and 403 compared to thetotal area of the body 301 contained within boxes 402, 403, and 404.According to one embodiment, the percent flashing (f) of the body 301can be at least about 1%. In another embodiment, the percent flashingcan be greater, such as at least about 2%, at least about 3%, at leastabout 5%, at least about 8%, at least about 10%, at least about 12%,such as at least about 15%, at least about 18%, or even at least about20%. Still, in a non-limiting embodiment, the percent flashing of thebody 301 can be controlled and may be not greater than about 45%, suchas not greater than about 40%, not greater than about 35%, not greaterthan about 30%, not greater than about 25%, not greater than about 20%,not greater than about 18%, not greater than about 15%, not greater thanabout 12%, not greater than about 10%, not greater than about 8%, notgreater than about 6%, or even not greater than about 4%. It will beappreciated that the percent flashing of the body 301 can be within arange between any of the above minimum and maximum percentages.Moreover, it will be appreciated that the above flashing percentages canbe representative of an average flashing percentage or a median flashingpercentage for a batch of shaped abrasive particles.

The percent flashing can be measured by mounting the shaped abrasiveparticle 300 on its side and viewing the body 301 at the side togenerate a black and white image, such as illustrated in FIG. 4. Asuitable program for such includes ImageJ software. The percentageflashing can be calculated by determining the area of the body 301 inthe boxes 402 and 403 compared to the total area of the body 301 asviewed at the side (total shaded area), including the area in the center404 and within the boxes. Such a procedure can be completed for asuitable sampling of particles to generate average, median, and/or andstandard deviation values.

FIGS. 12A-26 include illustrations of shaped abrasive particlesaccording to the embodiments herein. According to one embodiment, thebody of a shaped abrasive particle of the embodiments herein can have aparticular relationship between at least three grain features, includingtip sharpness, strength, and Shape Index. Without wishing to be tied toa particular theory, based on empirical studies it appears that aparticular interrelationship between certain grain features may exist,and by controlling the interrelationship of these grain features, theself-sharpening behavior of the shaped abrasive particle may bemodified, and improved, which may facilitate the formation of abrasivearticles having improved performance in terms of efficiency and life.

FIG. 12A includes a perspective view illustration of a shaped abrasiveparticle according to an embodiment. FIG. 12B includes a top viewillustration of a shaped abrasive particle according to an embodiment.As illustrated, the shaped abrasive particle 1200 can include a body1201 having an upper major surface 1203 (i.e., a first major surface)and a bottom major surface 1204 (i.e., a second major surface) oppositethe upper major surface 1203. The upper surface 1203 and the bottomsurface 1204 can be separated from each other by at least one sidesurface 1205, which may include one or more discrete side surfaceportions, including for example, discrete side surface portions 1206,1207, and 1208. The discrete side surface portions 1206-1208 may bejoined to each other at edges, including but not limited to, edges 1209and 1210. The edge 1209 can extend between an external corner 1211 ofthe upper major surface 1203 and an external corner 1212 of the bottommajor surface 1204. The edge 1210 can extend between an external corner1213 of the upper major surface 1203 and an external corner 1214 of thebottom major surface 1204.

As illustrated, the body 1201 of the shaped abrasive particle 1200 canhave a generally polygonal shape as viewed in a plane parallel to theupper surface 1203, and more particularly, a pentagonal two-dimensionalshape as viewed in the plane of the width and length of the body (i.e.,the top view as shown in FIG. 12B), having 5 external points or externalcorners. In particular, the body 1201 can have a length (L or Lmiddle)as shown in FIG. 12A, which may be measured as the dimension extendingfrom the external corner 1216 to a midpoint at the opposite edge 1217 ofthe body. Notably, in some embodiments, such as illustrated in FIG. 12A,the length can extend through a midpoint 1281 of the upper surface 1203of the body 1201, however, this may not necessarily be the case forevery embodiment. Moreover, the body 1201 can have a width (W), which isthe measure of the longest dimension of the body 1201 along a discreteside surface portion of the side surface 1205. The height of the bodymay be generally the distance between the upper major surface 1203 andthe bottom major surface 1204. As described in embodiments herein, theheight may vary in dimension at different locations of the body 1201,such as at the corners versus at the interior of the body 1201.

In particular instances, the body 1201 can be formed to have a primaryaspect ratio, which is a ratio expressed as width:length, having thevalues described in embodiments herein. Still, in certain embodiments,such as the shaped abrasive particle of the embodiment of FIG. 12A, thelength can be equal to or greater than the width, such that the primaryaspect ratio is at least about 1:1. In other instances, the body 1201can be formed such that the primary aspect ratio (w:l) can be at leastabout 1:1.5, such as at least about 1:2, at least about 1:4, or even atleast about 5:1. Still, in other instances, the abrasive particle 1200can be formed such that the body 1201 has a primary aspect ratio that isnot greater than about 1:10, such as not greater than 1:9, not greaterthan about 1:8, or even not greater than about 1:5. It will beappreciated that the body 1201 can have a primary aspect ratio within arange between any of the ratios noted above.

In addition to the primary aspect ratio, the abrasive particle 1200 canbe formed such that the body 1201 comprises a secondary aspect ratio,which can be defined as a ratio of length:height, wherein the height maybe an interior median height (Mhi) measured at the midpoint 1281. Incertain instances, the secondary aspect ratio can be at least about 1:1,such as at least about 2:1, at least about 4:1, or even at least about5:1. Still, in other instances, the abrasive particle 1200 can be formedsuch that the body 1201 has a secondary aspect ratio that is not greaterthan about 1:3, such as not greater than 1:2, or even not greater thanabout 1:1. It will be appreciated that the body 1201 can have asecondary aspect ratio within a range between any of the ratios notedabove, such as within a range between about 5:1 and about 1:1.

In accordance with another embodiment, the abrasive particle 1200 can beformed such that the body 1201 comprises a tertiary aspect ratio,defined by the ratio width:height, wherein the height may be an interiormedian height (Mhi). The tertiary aspect ratio of the body 1201 can beat least about 1:1, such as at least about 2:1, at least about 4:1, atleast about 5:1, or even at least about 6:1. Still, in other instances,the abrasive particle 1200 can be formed such that the body 1201 has atertiary aspect ratio that is not greater than about 3:1, such as notgreater than 2:1, or even not greater than about 1:1. It will beappreciated that the body 1201 can have a tertiary aspect ratio within arange between any of the ratios noted above, such as within a rangebetween about 6:1 and about 1:1.

According to one embodiment, the body 1201 of the shaped abrasiveparticle 1200 may be formed using any of the processes described herein.Notably, the body 1201 may be formed such that it has a particularinterrelationship of at least three grain features, including apredetermined strength, a predetermined tip sharpness, and apredetermined Shape Index. The tip sharpness of a shaped abrasiveparticle, which may be an average tip sharpness, may be measured bydetermining the largest radius of a best fit circle on an externalcorner of the body 1201. For example, turning to FIG. 12B, a top view ofthe upper major surface 1203 of the body 1201 is provided. For thecorner 1231, a best fit circle is overlaid on the image of the body 1201of the shaped abrasive particle 1201, and the radius of the best fitcircle relative to the curvature of the external corner 1231 defines thevalue of tip sharpness for the external corner 1231. The measurement maybe recreated for each external corner of the body 1201 to determine theaverage individual tip sharpness for a single shaped abrasive particle.Moreover, the measurement may be recreated on a suitable sample size ofshaped abrasive particles of a batch of shaped abrasive particles toderive the average batch tip sharpness. Any suitable computer program,such as ImageJ may be used in conjunction with an image (e.g., SEM imageor light microscope image) of suitable magnification to accuratelymeasure the best fit circle and the tip sharpness.

The shaped abrasive particles of the embodiments herein may have aparticular tip sharpness that facilitates formation of shaped abrasiveparticles with a particular sharpness, strength and Shape Index factor(i.e., 3SF). For example, the body of a shaped abrasive particle,according to an embodiment, can have a tip sharpness within a rangebetween not greater than about 80 microns and at least about 1 micron.Moreover, in certain instances, the body can have a tip sharpness of notgreater than about 78 microns, such as not greater than about 76microns, not greater than about 74 microns, not greater than about 72microns, not greater than about 70 microns, not greater than about 68microns, not greater than about 66 microns, not greater than about 64microns, not greater than about 62 microns, not greater than about 60microns, not greater than about 58 microns, not greater than about 56microns, not greater than about 54 microns, not greater than about 52microns, not greater than about 50 microns, not greater than about 48microns, not greater than about 46 microns, not greater than about 44microns, not greater than about 42 microns, not greater than about 40microns, not greater than about 38 microns, not greater than about 36microns, not greater than about 34 microns, not greater than about 32microns, not greater than about 30 microns, not greater than about 38microns, not greater than about 36 microns, not greater than about 34microns, not greater than about 32 microns, not greater than about 30microns, not greater than about 28 microns, not greater than about 26microns, not greater than about 24 microns, not greater than about 22microns, not greater than about 20 microns, not greater than about 18microns, not greater than about 16 microns, not greater than about 14microns, not greater than about 12 microns, not greater than about 10microns. In yet another non-limiting embodiment, the tip sharpness canbe at least about 2 microns, such as at least about 4 microns, at leastabout 6 microns, at least about 8 microns, at least about 10 microns, atleast about 12 microns, at least about 14 microns, at least about 16microns, at least about 18 microns, at least about 20 microns, at leastabout 22 microns, at least about 24 microns, at least about 26 microns,at least about 28 microns, at least about 30 microns, at least about 32microns, at least about 34 microns, at least about 36 microns, at leastabout 38 microns, at least about 40 microns, at least about 42 microns,at least about 44 microns, at least about 46 microns, at least about 48microns, at least about 50 microns, at least about 52 microns, at leastabout 54 microns, at least about 56 microns, at least about 58 microns,at least about 60 microns, at least about 62 microns, at least about 64microns, at least about 66 microns, at least about 68 microns, at leastabout 70 microns. It will be appreciated that the body can have a tipsharpness within a range between any of the minimum and maximum valuesnoted above.

As noted herein, another grain feature is the Shape Index. The ShapeIndex of the body 1201 can be described as a value of an outer radius ofa best-fit outer circle superimposed on the body as viewed in twodimensions of the plane of length and width (i.e., the upper majorsurface 1203 or the bottom major surface 1204) compared to an innerradius of the largest-best fit inner circle fitting entirely within thebody 1201 as viewed in the same dimensions of the plane of length andwidth of the body 1201. For example, turning to FIG. 12C, a top view ofthe shaped abrasive particle 1201 is provided with two circlessuperimposed on the illustration to demonstrate the calculation of ShapeIndex. A first circle is superimposed on the body of the shaped abrasiveparticle, which is a best-fit outer circle representing the smallestcircle that can be used to fit the entire perimeter of the body of theshaped abrasive particle within its boundaries. The outer circle has aradius (Ro). For shapes such as that illustrated in FIG. 12C, the outercircle may intersect the perimeter of the body at each of the fivecorners of the pentagon shape. However, it will be appreciated that forcertain irregular or complex shapes, the body may not fit uniformlywithin the circle such that each of the corners intersect the circle atequal intervals, but a best-fit, outer circle may be formed regardless.Any suitable computer program, such as ImageJ may be used in conjunctionwith an image of suitable magnification (e.g., SEM image or lightmicroscope image) to create the outer circle and measure the radius(Ro).

A second, inner circle can be superimposed on the image of a shapedabrasive grain, as illustrated in FIG. 12C, and is a best fit circlerepresenting the largest circle that can be placed entirely within theperimeter of the two dimensional shape of the body 1201 as viewed in theplane of the length and width of the body 1201. The inner circle canhave a radius (Ri). It will be appreciated that for certain irregular orcomplex shapes, the inner circle may not fit uniformly within the bodysuch that the perimeter of the circle contacts portions of the body atequal intervals, such as shown for the regular pentagon of FIG. 12C.However, a best-fit, inner circle may be formed regardless. Any suitablecomputer program, such as ImageJ may be used in conjunction with animage of suitable magnification (e.g., SEM image or light microscopeimage) to create the inner circle and measure the radius (Ri).

The Shape Index can be calculated by dividing the outer radius by theinner radius (i.e., Shape Index=Ri/Ro). For example, the body 1201 ofthe shaped abrasive particle 1200 of FIGS. 12A-12C has a Shape Index ofapproximately 0.81.

The shaped abrasive particles of the embodiments herein may have aparticular Shape Index that facilitates formation of shaped abrasiveparticles with a particular 3SF. For example, the body may have a ShapeIndex within a range between at least about 0.51 and not greater thanabout 0.99. More particularly, in one non-limiting embodiment, the bodyof the shaped abrasive particle can have a Shape Index of at least about0.52, such as at least about 0.53, at least about 0.54, at least about0.55, at least about 0.56, at least about 0.57, at least about 0.58, atleast about 0.59, at least about 0.60, at least about 0.61, at leastabout 0.62, at least about 0.63, at least about 0.64, at least about0.65, at least about 0.66, at least about 0.67, at least about 0.68, atleast about 0.69, at least about 0.70, at least about 0.71, at leastabout 0.72, at least about 0.73, at least about 0.74, at least about0.75, at least about 0.76, at least about 0.77, at least about 0.78, atleast about 0.79, at least about 0.80, at least about 0.81, at leastabout 0.82, at least about 0.83, at least about 0.84, at least about0.85, at least about 0.86, at least about 0.87, at least about 0.88, atleast about 0.89, at least about 0.90, at least about 0.91, at leastabout 0.92, at least about 0.93, at least about 0.94, at least about0.95. In still another non-limiting embodiment, the body can have aShape Index of not greater than about 0.98, such as not greater thanabout 0.97, not greater than about 0.96, not greater than about 0.95,not greater than about 0.94, not greater than about 0.93, not greaterthan about 0.92, not greater than about 0.91, not greater than about0.90, not greater than about 0.89, not greater than about 0.88, notgreater than about 0.87, not greater than about 0.86, not greater thanabout 0.85, not greater than about 0.84, not greater than about 0.83,not greater than about 0.82, not greater than about 0.81, not greaterthan about 0.80, not greater than about 0.79, not greater than about0.78, not greater than about 0.77, not greater than about 0.76, notgreater than about 0.75, not greater than about 0.74, not greater thanabout 0.73, not greater than about 0.72, not greater than about 0.71,not greater than about 0.70, not greater than about 0.69, not greaterthan about 0.68, not greater than about 0.67, not greater than about0.66, not greater than about 0.65, not greater than about 0.64, notgreater than about 0.63, not greater than about 0.62, not greater thanabout 0.61, not greater than about 0.60, not greater than about 0.59,not greater than about 0.58, not greater than about 0.57, not greaterthan about 0.56, not greater than about 0.55, not greater than about0.54. It will be appreciated that the body can have a Shape Index withina range between any of the minimum and maximum values noted above.

Moreover, as noted herein, the body 1201 may be formed to have aparticular strength. The strength of the body may be measured viaHertzian indentation. In this method the abrasive grains are glued on aslotted aluminum SEM sample mounting stub. The slots are approximately250 μm deep and wide enough to accommodate the grains in a row. Thegrains are polished in an automatic polisher using a series of diamondpastes, with the finest paste of 1 μm to achieve a final mirror finish.At the final step, the polished grains are flat and flush with thealuminum surface. The height of the polished grains is thereforeapproximately 250 μm. The metal stub is fixed in a metal support holderand indented with a steel spherical indenter using an MTS universal testframe. The crosshead speed during the test is 2 μm/s. The steel ballused as the indenter is 3.2 mm in diameter. The maximum indentation loadis the same for all grains, and the load at first fracture is determinedfrom the load displacement curve as a load drop. After indentation, thegrains are imaged optically to document the existence of the cracks andthe crack pattern.

Using the first load drop as the pop-in load of the first ring crack,the Hertzian strength can be calculated. The Hertzian stress field iswell defined and axiymmetrical. The stresses are compressive right underthe indenter and tensile outside a region defined by the radius of thecontact area. At low loads, the field is completely elastic. For asphere of radius R and an applied normal load of P, the solutions forthe stress field are readily found following the original Hertzianassumption that the contact is friction free.

The radius of the contact area a is given by:

$\begin{matrix}{a^{3} = \frac{3{PR}}{4E^{*}}} & (1)\end{matrix}$

Where

$\begin{matrix}{E^{*} = \left( {\frac{1 - v_{1}^{2}}{E_{1}} + \frac{1 - v_{2}^{2}}{E_{2}}} \right)^{- 1}} & (2)\end{matrix}$

and E* is a combination of the Elastic modulus E and the Poisson's ratioν for the indenter and sample material, respectively.

The maximum contact pressure is given by:

$\begin{matrix}{p_{0} = {\left( \frac{3P}{2\pi\mspace{14mu} a^{3}} \right) = \left( \frac{6{PE}^{*^{2}}}{\pi^{3}\mspace{14mu} R^{2}} \right)^{\frac{1}{3}}}} & (3)\end{matrix}$

The maximum shear stress is given by (assuming ν=0.3): τ₁=0.31, p₀, atR=0 and z=0.48 a

The Hertzian strength is the maximum tensile stress at the onset ofcracking and is calculated according to: σ_(r)=⅓ (1−2 ν) p₀, at R=a andz=0.

Using the first load drop as the load P in Eq. (3) the maximum tensilestress is calculated following the equation above, which is the value ofthe Hertzian strength for the specimen. In total, between 20 and 30individual shaped abrasive particle samples are tested for each grittype, and a range of Hertzian fracture stress is obtained. FollowingWeibull analysis procedures (as outlined in ASTM C1239), a Weibullprobability plot is generated, and the Weibull Characteristic strength(the scale value) and the Weibull modulus (the shape parameter) arecalculated for the distribution using the maximum likelihood procedure.

The shaped abrasive particles of the embodiments herein may have aparticular strength that facilitates formation of shaped abrasiveparticles with a particular 3SF. For example, the body of shapedabrasive particles of the embodiments herein can have a strength withina range between not greater than about 600 MPa and at least about 100MPa. This may be achieved using any of the compositions described in theembodiments herein, including but not limited to, a single ceramiccomposition, a doped ceramic composition, or a composite composition.According to a particular embodiment, the strength of the body may benot greater than about 590 MPa, such as not greater than about 580 MPa,not greater than about 570 MPa, not greater than about 560 MPa, notgreater than about 550 MPa, not greater than about 540 MPa, not greaterthan about 530 MPa, not greater than about 520 MPa, not greater thanabout 510 MPa, not greater than about 500 MPa, not greater than about490 MPa, not greater than about 480 MPa, not greater than about 470 MPa,not greater than about 460 MPa, not greater than about 450 MPa, notgreater than about 440 MPa, not greater than about 430 MPa, not greaterthan about 420 MPa, not greater than about 410 MPa, not greater thanabout 400 MPa, not greater than about 390 MPa, not greater than about380 MPa, not greater than about 370 MPa, not greater than about 360 MPa,not greater than about 350 MPa, not greater than about 340 MPa, notgreater than about 330 MPa, not greater than about 320 MPa, not greaterthan about 310 MPa, not greater than about 300 MPa, not greater thanabout 290 MPa, not greater than about 280 MPa, not greater than about270 MPa, not greater than about 260 MPa, not greater than about 250 MPa,not greater than about 240 MPa, not greater than about 230 MPa, notgreater than about 220 MPa, not greater than about 210 MPa, or even notgreater than about 200 MPa. In yet another non-limiting embodiment, thestrength of the body may be at least about 110 MPa, such as at leastabout 120 MPa, at least about 130 MPa, at least about 140 MPa, at leastabout 150 MPa, at least about 160 MPa, at least about 170 MPa, at leastabout 180 MPa, at least about 190 MPa, at least about 200 MPa, at leastabout 210 MPa, at least about 220 MPa, at least about 230 MPa, at leastabout 240 MPa, at least about 250 MPa, at least about 260 MPa, at leastabout 270 MPa, at least about 280 MPa, at least about 290 MPa, at leastabout 300 MPa, at least about 310 MPa, at least about 320 MPa, at leastabout 330 MPa, at least about 340 MPa, at least about 350 MPa, at leastabout 360 MPa, at least about 370 MPa, at least about 380 MPa, at leastabout 390 MPa, at least about 400 MPa, at least about 410 MPa, at leastabout 420 MPa, at least about 430 MPa, at least about 440 MPa, at leastabout 450 MPa, at least about 460 MPa, at least about 470 MPa, at leastabout 480 MPa, at least about 490 MPa, or even at least about 500. Itwill be appreciated that the strength of the body may be within a rangebetween any of the minimum and maximum values noted above.

According to one aspect, empirical studies of shaped abrasive particleshave indicated that by controlling particular grain features of tipsharpness, strength, and Shape Index with respect to each other, thegrinding behavior (e.g., the self-sharpening behavior) of the shapedabrasive particles can be modified. Notably, the forming process can beundertaken in a manner such that the interrelationship of the grainfeatures of tip sharpness, Shape Index, and strength of the body areselected and controlled in a predetermined manner to influence thegrinding performance (e.g., self-sharpening behavior) of the shapedabrasive particle. For example, in one embodiment, the method of formingthe shaped abrasive particle can include selecting a material having apredetermined strength and forming the body of the shaped abrasiveparticle with a predetermined tip sharpness and predetermined ShapeIndex based upon the predetermined strength. That is, a material forforming the shaped abrasive particle may first be selected, such thatthe body will have a predetermined strength, and thereafter the grainfeatures of a predetermined tip sharpness and predetermined Shape Indexmay be selected and controlled based on the predetermined strength, suchthat the shaped abrasive particle may have improved performance overconventional shaped abrasive particles.

In still another embodiment, the method of forming the shaped abrasiveparticle can include selecting a material having a predetermined ShapeIndex and forming the body of the shaped abrasive particle with apredetermined tip sharpness and predetermined strength based upon thepredetermined Shape Index. That is, a shape of the body of the shapedabrasive particle may first be selected, and thereafter the grainfeatures of a predetermined tip sharpness and predetermined strength ofthe body may be selected and controlled based on the predetermined ShapeIndex, such that the shaped abrasive particle can have improvedperformance over conventional shaped abrasive particles.

In yet another approach, a method of forming a shaped abrasive particlecan include selecting a predetermined tip sharpness of a body of theshaped abrasive particle. After predetermining the tip sharpness of thebody, the Shape Index and the strength of the body may be selected andcontrolled based upon the predetermined tip sharpness. Such a processmay facilitate formation of a shaped abrasive particle having improvedperformance over conventional shaped abrasive particles.

In yet another embodiment, the method of forming the shaped abrasiveparticle can include selecting a material having a predetermined height,which may be an average height, an interior height, or height at an edgeor tip of the body, and forming the body of the shaped abrasive particlewith a predetermined tip sharpness, predetermined strength, andpredetermined Shape Index based on the predetermined height. That is, aheight of the body of the shaped abrasive particle may first beselected, and thereafter the grain features of a predetermined tipsharpness, strength, and Shape Index of the body may be selected andcontrolled based on the predetermined height, such that the shapedabrasive particle can have improved performance over conventional shapedabrasive particles.

Moreover, through empirical studies, it has been found that theperformance of the shaped abrasive particle may be initially predictedby the interrelationship of the tip sharpness, strength, and ShapeIndex, which may be evaluated based upon a sharpness-shape-strengthfactor (3SF) according to the formula: 3SF=[(S*R*B²)/2500], wherein “S”represents the strength of the body (in MPa), R represents the tipsharpness of the body (in microns), and “B” represents the Shape Indexof the body. The 3SF formula is intended to provide an initialprediction of the effectiveness of grinding behavior of the particlebased upon the interrelationship of the grain features. It should benoted that other factors, such as aspects of the abrasive article inwhich the shaped abrasive particle is integrated, may influence thebehavior of the particle.

In accordance with one embodiment, the body of the shaped abrasiveparticle may have a particular 3SF value within a range between at leastabout 0.7 and not greater than about 1.7. In at least one embodiment,the body can have a 3SF of at least about 0.72, such as at least about0.75, at least about 0.78, at least about 0.8, at least about 0.82, atleast about 0.85, at least about 0.88, at least about 0.90, at leastabout 0.92, at least about 0.95, or even at least about 0.98. In yetanother instance, the body can have a 3SF of not greater than about1.68, such as not greater than about 1.65, not greater than about 1.62,not greater than about 1.6, not greater than about 1.58, not greaterthan about 1.55, not greater than about 1.52, not greater than about1.5, not greater than about 1.48, not greater than about 1.45, notgreater than about 1.42, not greater than about 1.4, not greater thanabout 1.38, not greater than about 1.35, not greater than about 1.32,not greater than about 1.3, not greater than about 1.28, not greaterthan about 1.25, not greater than about 1.22, not greater than about1.2, not greater than about 1.18, not greater than about 1.15, notgreater than about 1.12, not greater than about 1.1. It will beappreciated that the body can have a 3SF within a range between any ofthe minimum and maximum values noted above.

In addition to the foregoing grain features and 3SF values of theembodiments herein, in certain instances, the height of the grain may bean additional or alternative grain feature that may be interrelated tocertain grain features described herein. In particular, the height ofthe grain may be controlled with respect to any of the grain features(e.g., strength and tip sharpness) to facilitate improved grindingperformance of the shaped abrasive particles and abrasive articles usingsuch shaped abrasive particles. Notably, the shaped abrasive particlesof the embodiments herein can have a particular height, which may beinterrelated to certain grain features, such that stresses encounteredduring grinding may be distributed throughout the body in a manner tofacilitate improved self-sharpening behavior. According to oneembodiment, the body of the shaped abrasive particles can have a height(h) within a range between about 70 microns and about 500 microns, suchas within a range between about 175 microns to about 350 microns, suchas between about 175 microns and about 300 microns, or even within arange between about 200 microns and about 300 microns.

The shaped abrasive particles of the embodiments herein having theparticular grain features and 3SF can have any of the other features ofthe embodiments described herein. In one aspect, the body 1201 of theshaped abrasive particle can have a particular composition. For example,the body 1201 may include a ceramic material, such as a polycrystallineceramic material, and more particularly an oxide. The oxide may include,for example alumina. In certain instances, the body may include amajority content of alumina, such as at least about 95 wt % alumina forthe total weight of the body, or such as at least about 95.1 wt %, atleast about 95.2 wt %, at least about 95.3 wt %, at least about 95.4 wt%, at least about 95.5 wt %, at least about 95.6 wt %, at least about95.7 wt %, at least about 95.8 wt %, at least about 95.9 wt %, at leastabout 96 wt %, at least about 96.1 wt %, at least about 96.2 wt %, atleast about 96.3 wt %, at least about 96.4 wt %, at least about 96.5 wt%, at least about 96.6 wt %, at least about 96.7 wt %, at least about96.8 wt %, at least about 96.9 wt %, at least about 97 wt %, at leastabout 97.1 wt %, at least about 97.2 wt %, at least about 975.3 wt %, atleast about 97.4 wt %, or even at least about 97.5 wt % alumina for thetotal weight of the body. Still, in another non-limiting embodiment, thebody 1201 may include a content of alumina not greater than about 99.5wt %, such as not greater than about 99.4 wt %, not greater than about99.3 wt %, not greater than about 99.2 wt %, not greater than about 99.1wt %, not greater than about 99 wt %, not greater than about 98.9 wt %,not greater than about 98.8 wt %, not greater than about 98.7 wt %, notgreater than about 98.6 wt %, not greater than about 98.5 wt %, notgreater than about 98.4 wt %, not greater than about 98.3 wt %, notgreater than about 98.2 wt %, not greater than about 98.1 wt %, notgreater than about 98 wt %, not greater than about 97.9 wt %, notgreater than about 97.8 wt %, not greater than about 97.7 wt %, notgreater than about 97.6 wt %, or even not greater than about 97.5 wt %alumina for the total weight of the body 1201. It will be appreciatedthat the body 1201 may include a content of alumina within a rangebetween any of the minimum and maximum values noted above. Moreover, inat least one embodiment, the body may consist essentially of alumina.

As noted in embodiments herein, the body of the shaped abrasiveparticles may be formed to include certain additives. The additives canbe non-organic species, including but not limited to an oxide, a metalelement, a rare-earth element, and a combination thereof. In oneparticular instance, the additive may be a dopant material, which may bepresent in a particular minor amount sufficient to affect themicrostructure of the material, but not necessarily present in a traceamount or less. The dopant material may include an element selected fromthe group consisting of an alkali element, an alkaline earth element, arare earth element, a transition metal element, and a combinationthereof. More particularly, the dopant material can be an elementselected from the group consisting of hafnium, zirconium, niobium,tantalum, molybdenum, vanadium, lithium, sodium, potassium, magnesium,calcium, strontium, barium, scandium, yttrium, lanthanum, cesium,praseodymium, chromium, cobalt, iron, germanium, manganese, nickel,titanium, zinc, and a combination thereof. In still a more particularembodiment, the dopant material may include a magnesium-containingspecies, including for example, but not limited to, and may be magnesiumoxide (MgO).

According to one embodiment, the magnesium-containing species can be acompound including magnesium and at least one other element. In at leastone embodiment, the magnesium-containing compound can include an oxidecompound, such that the magnesium-containing species includes magnesiumand oxygen. In yet another embodiment, the magnesium-containing speciescan include aluminum, and more particularly may be a magnesium aluminatespecies. For example, in certain instances, the magnesium-containingspecies can be a spinel material. The spinel material may bestoichiometric or non-stoichiometric spinel.

The magnesium-containing species may be a distinct phase of materialformed in the body as compared to another primary phase, including forexample, an alumina phase. The magnesium-containing species may bepreferentially disposed at the grain boundaries of the primary phase(e.g., alumina grains). In still other instances, themagnesium-containing species may be primarily and uniformly dispersedthroughout the volume of the grains of the primary phase.

The magnesium-containing species may be a strength-altering material.For example, in at least one embodiment, the addition of themagnesium-containing species can be configured to reduce the strength ofthe body compared to a body that does not include themagnesium-containing species.

Certain compositions of the shaped abrasive particles of the embodimentscan include a particular content of magnesium oxide. For example, thebody 1201 may include a content of the magnesium-containing species ofat least about 0.5 wt %, such as at least about 0.6 wt %, at least about0.7 wt %, at least about 0.8 wt %, at least about 0.9 wt %, at leastabout 1 wt %, at least about 1.1 wt %, at least about 1.2 wt %, at leastabout 1.3 wt %, at least about 1.4 wt %, at least about 1.5 wt %, atleast about 1.6 wt %, at least about 1.7 wt %, at least about 1.8 wt %,at least about 1.9 wt %, at least about 2 wt %, at least about 2.1 wt %,at least about 2.2 wt %, at least about 2.3 wt %, at least about 2.4 wt%, or even at least about 2.5 wt % for the total weight of the body1201. In still another non-limiting embodiment, the body 1201 mayinclude a content of the magnesium-containing species of not greaterthan about 8 wt %, not greater than about 7 wt %, not greater than about6 wt %, not greater than about 5 wt %, not greater than about 4.9 wt %,not greater than about 4.8 wt %, not greater than about 4.7 wt %, notgreater than about 4.6 wt %, not greater than about 4.5 wt %, notgreater than about 4.4 wt %, not greater than about 4.3 wt %, notgreater than about 4.2 wt %, not greater than about 4.1 wt %, notgreater than about 4 wt %, not greater than about 3.9 wt %, not greaterthan about 3.8 wt %, not greater than about 3.7 wt %, not greater thanabout 3.6 wt %, not greater than about 3.5 wt %, not greater than about3.4 wt %, not greater than about 3.3 wt %, not greater than about 3.2 wt%, not greater than about 3.1 wt %, not greater than about 3 wt %, notgreater than about 2.9 wt %, not greater than about 2.8 wt %, notgreater than about 2.7 wt %, not greater than about 2.6 wt %, notgreater than about 2.5 wt %. It will be appreciated that the content ofthe magnesium-containing species within the body may be within a rangebetween any of the minimum and maximum values noted above. Furthermore,in at least one embodiment, the body 1201 may consist essentially ofalumina (Al₂O₃) and the magnesium-containing species (e.g., MgO and/or amagnesium aluminate).

Moreover, as noted herein the body of a shaped abrasive particle of anyof the embodiments herein may be formed of a polycrystalline materialincluding grains, which may be made of materials such as nitrides,oxides, carbides, borides, oxynitrides, diamond, and a combinationthereof. Further, the body 1201 can be essentially free of an organicmaterial, essentially free of rare earth elements, and essentially freeof iron. Being essentially free is understood to mean that the body isformed in a manner to exclude such materials, but the body may notnecessarily be completely free of such materials as they may be presentin trace amounts or less.

FIG. 13A includes a top view of a shaped abrasive particle according toan embodiment. The shaped abrasive particle 1300 can have a body 1301having the features of other shaped abrasive particles of embodimentsherein, including an upper major surface 1303 and a bottom major surface(not shown) opposite the upper major surface 1303. The upper majorsurface 1303 and the bottom major surface can be separated from eachother by at least one side surface 1304, which may include one or morediscrete side surface sections. According to one embodiment, the body1301 can be defined as an irregular hexagon, wherein the body has ahexagonal (i.e., six-sided) two dimensional shape as viewed in the planeof a length and a width of the body 1301, and wherein at least two ofthe sides, such as sides 1305 and 1306, have a different length withrespect to each other. Notably, the length of the sides is understoodherein to refer to the width of the body 1301 and the length of the bodyis the greatest dimension extending through the midpoint of the body1301. Moreover, as illustrated, none of the sides are parallel to eachother. And furthermore, while not illustrated, any of the sides may havea curvature to them, including a concave curvature wherein the sides maycurve inwards toward the midpoint of the body 1301 between cornersjoining two sides.

According to a more particular embodiment, the body 1301 can have anoblique, truncated shape as viewed top-down. In such embodiments, theside surface can include a first side section 1305 and a first obliqueside section 1306, which can be joined to each other at a first obliquecorner 1307 defining a first oblique corner angle Ao1. Notably, thefirst side section 1305 and the first oblique side section 1306 can bejoined to each other in a particular manner such that the first obliqueangle Ao1 can be an obtuse angle. In more particular instances, thefirst oblique angle Ao1 can have an obtuse value of at least about 92degrees, such as at least about 94 degrees, at least about 96 degrees,at least about 98 degrees, at least about 100 degrees, at least about102 degrees, at least about 104 degrees, at least about 106 degrees, atleast about 108 degrees, at least about 110 degrees, at least about 112degrees, at least about 124 degrees, at least about 126 degrees, atleast about 128 degrees, at least about 120 degrees, at least about 122degrees, at least about 124 degrees, at least about 126 degrees, atleast about 128 degrees, at least about 130 degrees, at least about 132degrees, at least about 134 degrees, at least about 136 degrees, atleast about 138 degrees, or even at least about 140 degrees. Still, inat least one non-limiting embodiment, the first oblique angle Ao1 can bean obtuse angle having a value of not greater than about 176 degrees,such as not greater than about 174 degrees, not greater than about 172degrees, not greater than about 170 degrees, not greater than about 168degrees, not greater than about 166 degrees, not greater than about 164degrees, not greater than about 162 degrees, not greater than about 160degrees, not greater than about 158 degrees, not greater than about 156degrees, not greater than about 154 degrees, not greater than about 152degrees, not greater than about 150 degrees, not greater than about 148degrees, not greater than about 146 degrees, not greater than about 144degrees, not greater than about 142 degrees, or even not greater thanabout 140 degrees. It will be appreciated that the first oblique angleAo1 can have a value within a range between any of the minimum andmaximum values noted above.

As further illustrated in the embodiment of FIG. 13A, the shapedabrasive particle can have a body 1301, wherein the first side section1305 can have a first side section length (Lss1) and the first obliqueside section 1306 can have a length (Los1). In certain instances, thelength of the first oblique side section (Los1) can be different thanthe length of the first side section (Lss1). For example, in certainembodiments, the length of the first oblique side section (Los1) can begreater than the length of the first side section (Lss1) (i.e.,Los1>Lss1). In another embodiment, the length of the first side section(Lss1) can be greater than the length of the first oblique side section(Los1) (i.e., Lss1>Los1).

In at least one particular instance, the relationship between the lengthof the first oblique side section (Los1) and the length of the firstside section (Lss1) can define a length factor (Los1/Lss1) that mayfacilitate improved performance of the shaped abrasive particle 1300.For example, the length factor (Los1/Lss1) can be not greater than about1, such as not greater than about 0.95, not greater than about 0.9, notgreater than about 0.85, not greater than about 0.8, not greater thanabout 0.75, not greater than about 0.7, not greater than about 0.65, notgreater than about 0.6, not greater than about 0.55, not greater thanabout 0.5, not greater than about 0.45, not greater than about 0.4, notgreat not greater than about 0.35, not greater than about 0.3, notgreater than about 0.35, not greater than about 0.3, not greater thanabout 0.25, not greater than about 0.2, not greater than about 0.15, notgreater than about 0.1, or even not greater than about 0.05. For yetanother non-limiting embodiment, the length factor (Los1/Lss1) can be atleast about 0.05, such as at least about 0.1, at least about 0.15, atleast about 0.2, at least about 0.25, at least about 0.3, at least about0.35, at least about 0.4, at least about 0.45, at least about 0.5, atleast about 0.55, at least about 0.6, at least about 0.65, at leastabout 0.7, at least about 0.75, at least about 0.8, at least about 0.85,at least about 0.9, or even at least about 0.95. It will be appreciatedthat the length factor (Los1/Lss1) can be within a range between any ofthe minimum and maximum values noted above.

According to an alternative embodiment, the relationship between thelength of the first oblique side section (Los1) and the length of thefirst side section (Lss1) can define a length factor (Lss1/Los1) thatmay facilitate improved performance of the shaped abrasive particle1300. For example, the length factor (Lss1/Los1) can be not greater thanabout 1, such as not greater than about 0.95, not greater than about0.9, not greater than about 0.85, not greater than about 0.8, notgreater than about 0.75, not greater than about 0.7, not greater thanabout 0.65, not greater than about 0.6, not greater than about 0.55, notgreater than about 0.5, not greater than about 0.45, not greater thanabout 0.4, not great not greater than about 0.35, not greater than about0.3, not greater than about 0.35, not greater than about 0.3, notgreater than about 0.25, not greater than about 0.2, not greater thanabout 0.15, not greater than about 0.1, or even not greater than about0.05. For yet another non-limiting embodiment, the length factor(Lss1/Los1) can be at least about 0.05, such as at least about 0.1, atleast about 0.15, at least about 0.2, at least about 0.25, at leastabout 0.3, at least about 0.35, at least about 0.4, at least about 0.45,at least about 0.5, at least about 0.55, at least about 0.6, at leastabout 0.65, at least about 0.7, at least about 0.75, at least about 0.8,at least about 0.85, at least about 0.9, or even at least about 0.95. Itwill be appreciated that the length factor (Lss1/Los1) can be within arange between any of the minimum and maximum values noted above.

As further illustrated, the second side section 1311 and the firstoblique side section 1306 can be joined to each other and define a firstexternal corner 1309. The first external corner 1309 can define a firstexternal corner angle Aec1. In certain instances, the first externalcorner angle Aec1 can be different than a value of the first obliqueangle Ao1. In at least one embodiment, the first external corner angleAec1 can be less than the value of the first oblique angle Ao1.

The first external corner angle Aec1 may be formed to have a particularvalue that may faciltiate improved performance of the shaped abrasiveparticle. For example, the first external corner angle Aec1 may be notgreater than about 130 degrees, such as not greater than about 125degrees, not greater than about 120 degrees, not greater than about 115degrees, not greater than about 110 degrees, not greater than about 105degrees, not greater than about 100 degrees, not greater than about 95degrees, not greater than about 94 degrees, or even not greater thanabout 93 degrees. Still, in at least one non-limiting embodiment, thefirst external corner angle Aec1 can be at least about 50 degrees, suchas at least about 55 degrees, at least about 60 degrees, at least about65 degrees, at least about 70 degrees, at least about 80 degrees, oreven at least about 85 degrees. It will be appreciated that the firstexternal corner angle Aec1 can have a value within a range between anyof the minimum and maximum values noted above. In one particularembodiment, the first external corner angle Aec1 can be substantiallyperpendicular.

The first external corner angle Aec1 and the first oblique angle Ao1 maybe formed to have a particular relationship, which may be described as afirst angle factor (Aec1/Ao1) having a particular value that mayfacilitate improved performance of the shaped abrasive particle 1300.For example, the first angle factor (Aec1/Ao1) may be not greater thanabout 1, such as not greater than about 0.95, not greater than about0.9, not greater than about 0.85, not greater than about 0.8, notgreater than about 0.75, not greater than about 0.7, not greater thanabout 0.65, not greater than about 0.6, not greater than about 0.55, notgreater than about 0.5, not greater than about 0.45, not greater thanabout 0.4, not great not greater than about 0.35, not greater than about0.3, not greater than about 0.35, not greater than about 0.3, notgreater than about 0.25, not greater than about 0.2, not greater thanabout 0.15, not greater than about 0.1, or even not greater than about0.05. In yet another embodiment, the first angle factor (Aec1/Ao1) maybe at least about 0.05, such as at least about 0.1, at least about 0.15,at least about 0.2, at least about 0.25, at least about 0.3, at leastabout 0.35, at least about 0.4, at least about 0.45, at least about 0.5,at least about 0.55, at least about 0.6, at least about 0.65, at leastabout 0.7, at least about 0.75, at least about 0.8, at least about 0.85,at least about 0.9, or even at least about 0.95. It will be appreciatedthat the first angle factor (Aec1/Ao1) may be within a range between anyof the minimum and maximum values noted above.

As further illustrated, the body 1301 can have a side surface 1304including a second side section 1311 and a second oblique side section1312, which can be joined to each other at a second oblique angle Ao2.Notably, the second side section 1311 and the second oblique sidesection 1312 can be joined to each other in a particular manner suchthat the second oblique angle Ao2 can be an obtuse angle. In moreparticular instances, the second oblique angle Ao2 can have an obtusevalue of at least about 92 degrees, such as at least about 94 degrees,at least about 96 degrees, at least about 98 degrees, at least about 100degrees, at least about 102 degrees, at least about 104 degrees, atleast about 106 degrees, at least about 108 degrees, at least about 110degrees, at least about 112 degrees, at least about 124 degrees, atleast about 126 degrees, at least about 128 degrees, at least about 120degrees, at least about 122 degrees, at least about 124 degrees, atleast about 126 degrees, at least about 128 degrees, at least about 130degrees, at least about 132 degrees, at least about 134 degrees, atleast about 136 degrees, at least about 138 degrees, or even at leastabout 140 degrees. Still, in at least one non-limiting embodiment, thesecond oblique angle Ao2 can be an obtuse angle having a value of notgreater than about 176 degrees, such as not greater than about 174degrees, not greater than about 172 degrees, not greater than about 170degrees, not greater than about 168 degrees, not greater than about 166degrees, not greater than about 164 degrees, not greater than about 162degrees, not greater than about 160 degrees, not greater than about 158degrees, not greater than about 156 degrees, not greater than about 154degrees, not greater than about 152 degrees, not greater than about 150degrees, not greater than about 148 degrees, not greater than about 146degrees, not greater than about 144 degrees, not greater than about 142degrees, or even not greater than about 140 degrees. It will beappreciated that the second oblique angle Ao2 can have a value within arange between any of the minimum and maximum values noted above.

Moreover, the shaped abrasive particle can have a body 1301, wherein thesecond side section 1311 can have a second side section length (Lss2)and the second oblique side section 1312 can have a length (Los2). Incertain instances, the length of the second oblique side section (Los2)can be different than the length of the second side section (Lss2). Forexample, in certain embodiments, the length of the second oblique sidesection (Los2) can be greater than the length of the second side section(Lss2) (i.e., Los2>Lss2). In another embodiment, the length of thesecond side section (Lss2) can be greater than the length of the secondoblique side section (Los2) (i.e., Lss2>Los2).

In at least one aspect, the relationship between the length of thesecond oblique side section (Los2) and the length of the second sidesection (Lss2) can define a length factor (Los2/Lss2) that mayfacilitate improved performance of the shaped abrasive particle 1300.For example, the length factor (Los2/Lss2) can be not greater than about1, such as not greater than about 0.95, not greater than about 0.9, notgreater than about 0.85, not greater than about 0.8, not greater thanabout 0.75, not greater than about 0.7, not greater than about 0.65, notgreater than about 0.6, not greater than about 0.55, not greater thanabout 0.5, not greater than about 0.45, not greater than about 0.4, notgreat not greater than about 0.35, not greater than about 0.3, notgreater than about 0.35, not greater than about 0.3, not greater thanabout 0.25, not greater than about 0.2, not greater than about 0.15, notgreater than about 0.1, or even not greater than about 0.05. For yetanother non-limiting embodiment, the length factor (Los2/Lss2) can be atleast about 0.05, such as at least about 0.1, at least about 0.15, atleast about 0.2, at least about 0.25, at least about 0.3, at least about0.35, at least about 0.4, at least about 0.45, at least about 0.5, atleast about 0.55, at least about 0.6, at least about 0.65, at leastabout 0.7, at least about 0.75, at least about 0.8, at least about 0.85,at least about 0.9, or even at least about 0.95. It will be appreciatedthat the length factor (Los2/Lss2) can be within a range between any ofthe minimum and maximum values noted above.

In an alternative embodiment, the relationship between the length of thesecond oblique side section (Los2) and the length of the second sidesection (Lss2) can define a length factor (Lss2/Los2) that mayfacilitate improved performance of the shaped abrasive particle 1300.For example, the length factor (Lss2/Los2) can be not greater than about1, such as not greater than about 0.95, not greater than about 0.9, notgreater than about 0.85, not greater than about 0.8, not greater thanabout 0.75, not greater than about 0.7, not greater than about 0.65, notgreater than about 0.6, not greater than about 0.55, not greater thanabout 0.5, not greater than about 0.45, not greater than about 0.4, notgreat not greater than about 0.35, not greater than about 0.3, notgreater than about 0.35, not greater than about 0.3, not greater thanabout 0.25, not greater than about 0.2, not greater than about 0.15, notgreater than about 0.1, or even not greater than about 0.05. For yetanother non-limiting embodiment, the length factor (Lss2/Los2) can be atleast about 0.05, such as at least about 0.1, at least about 0.15, atleast about 0.2, at least about 0.25, at least about 0.3, at least about0.35, at least about 0.4, at least about 0.45, at least about 0.5, atleast about 0.55, at least about 0.6, at least about 0.65, at leastabout 0.7, at least about 0.75, at least about 0.8, at least about 0.85,at least about 0.9, or even at least about 0.95. It will be appreciatedthat the length factor (Lss2/Los2) can be within a range between any ofthe minimum and maximum values noted above.

Additionally, the length of the second side section (Lss2) relative tothe length of the first side section (Lss1) may be controlled tofacilitate improved performance of the shaped abrasive particle 1300. Inone embodiment, Lss2 is different compared to Lss1. For example, Lss2can be greater than Lss1. In still other embodiments, Lss2 can be lessthan Lss1. For yet another embodiment, such as illustrated in FIG. 13A,Lss1 and Lss2 can be essentially the same compared to each other.

Moreover, the length of the second oblique side section (Los2) relativeto the length of the first oblique side section (Los1) may be controlledto facilitate improved performance of the shaped abrasive particle 1300.In one embodiment, Los2 is different compared to Los1. For example, Los2can be greater than Los1. In still other embodiments, Los2 can be lessthan Los1. For yet another embodiment, such as illustrated in FIG. 13A,Los1 and Los2 can be essentially the same compared to each other.

As further illustrated, the side surface 1304 can include a third sidesection 1317 joined to the second oblique side section 1312 to define asecond external corner 1315. The second external corner 1315 can definea second external corner angle Aec2. In certain instances, the secondexternal corner angle Aec2 can be different than a value of the secondoblique angle Ao2. In at least one embodiment, the second externalcorner angle Aec2 can be less than the value of the second oblique angleAo2.

The second external corner angle Aec2 can be formed to have a particularvalue that may faciltiate improved performance of the shaped abrasiveparticle. For example, the second external corner angle Aec2 may be notgreater than about 130 degrees, such as not greater than about 125degrees, not greater than about 120 degrees, not greater than about 115degrees, not greater than about 110 degrees, not greater than about 105degrees, not greater than about 100 degrees, not greater than about 95degrees, not greater than about 94 degrees, or even not greater thanabout 93 degrees. Still, in at least one non-limiting embodiment, thesecond external corner angle Aec2 can be at least about 50 degrees, suchas at least about 55 degrees, at least about 60 degrees, at least about65 degrees, at least about 70 degrees, at least about 80 degrees, oreven at least about 85 degrees. It will be appreciated that the secondexternal corner angle Aec2 can have a value within a range between anyof the minimum and maximum values noted above. In one particularembodiment, the second external corner angle Aec2 can be substantiallyperpendicular.

The second external corner angle Aec2 and the second oblique angle Ao2may be formed to have a particular relationship with respect to eachother, which may be described as a second angle factor (Aec2/Ao2) havinga particular value that may facilitate improved performance of theshaped abrasive particle 1300. For example, the second angle factor(Aec2/Ao2) may be not greater than about 1, such as not greater thanabout 0.95, not greater than about 0.9, not greater than about 0.85, notgreater than about 0.8, not greater than about 0.75, not greater thanabout 0.7, not greater than about 0.65, not greater than about 0.6, notgreater than about 0.55, not greater than about 0.5, not greater thanabout 0.45, not greater than about 0.4, not great not greater than about0.35, not greater than about 0.3, not greater than about 0.35, notgreater than about 0.3, not greater than about 0.25, not greater thanabout 0.2, not greater than about 0.15, not greater than about 0.1, oreven not greater than about 0.05. In yet another embodiment, the secondangle factor (Aec2/Ao2) may be at least about 0.05, such as at leastabout 0.1, at least about 0.15, at least about 0.2, at least about 0.25,at least about 0.3, at least about 0.35, at least about 0.4, at leastabout 0.45, at least about 0.5, at least about 0.55, at least about 0.6,at least about 0.65, at least about 0.7, at least about 0.75, at leastabout 0.8, at least about 0.85, at least about 0.9, or even at leastabout 0.95. It will be appreciated that the second angle factor(Aec2/Ao2) may be within a range between any of the minimum and maximumvalues noted above.

As further illustrated, the body 1301 can have a side surface 1304including the third side section 1317 and a third oblique side section1319, which can be joined to each other at a third oblique corner 1318defining a third oblique angle Ao3. Notably, the third side section 1317and the third oblique side section 1319 can be joined to each other in aparticular manner such that the third oblique angle Ao3 can be an obtuseangle. In more particular instances, the third oblique angle Ao3 canhave an obtuse value of at least about 92 degrees, such as at leastabout 94 degrees, at least about 96 degrees, at least about 98 degrees,at least about 100 degrees, at least about 102 degrees, at least about104 degrees, at least about 106 degrees, at least about 108 degrees, atleast about 110 degrees, at least about 112 degrees, at least about 124degrees, at least about 126 degrees, at least about 128 degrees, atleast about 120 degrees, at least about 122 degrees, at least about 124degrees, at least about 126 degrees, at least about 128 degrees, atleast about 130 degrees, at least about 132 degrees, at least about 134degrees, at least about 136 degrees, at least about 138 degrees, or evenat least about 140 degrees. Still, in at least one non-limitingembodiment, the third oblique angle Ao3 can be an obtuse angle having avalue of not greater than about 176 degrees, such as not greater thanabout 174 degrees, not greater than about 172 degrees, not greater thanabout 170 degrees, not greater than about 168 degrees, not greater thanabout 166 degrees, not greater than about 164 degrees, not greater thanabout 162 degrees, not greater than about 160 degrees, not greater thanabout 158 degrees, not greater than about 156 degrees, not greater thanabout 154 degrees, not greater than about 152 degrees, not greater thanabout 150 degrees, not greater than about 148 degrees, not greater thanabout 146 degrees, not greater than about 144 degrees, not greater thanabout 142 degrees, or even not greater than about 140 degrees. It willbe appreciated that the third oblique angle Ao3 can have a value withina range between any of the minimum and maximum values noted above.

In certain instances, the shaped abrasive particle can have a body 1301,wherein the third side section 1317 can have a third side section length(Lss3) and the third oblique side section 1319 can have a length (Los3).Moreover, the length of the third oblique side section (Los3) can bedifferent than the length of the third side section (Lss3). For example,in certain embodiments, the length of the third oblique side section(Los3) can be greater than the length of the third side section (Lss3)(i.e., Los3>Lss3). In another embodiment, the length of the third sidesection (Lss3) can be greater than the length of the third oblique sidesection (Los3) (i.e., Lss3>Los3).

In at least one aspect, the relationship between the length of the thirdoblique side section (Los3) and the length of the third side section(Lss3) can define a length factor (Los3/Lss3), which may facilitateimproved performance of the shaped abrasive particle 1300. For example,the length factor (Los3/Lss3) can be not greater than about 1, such asnot greater than about 0.95, not greater than about 0.9, not greaterthan about 0.85, not greater than about 0.8, not greater than about0.75, not greater than about 0.7, not greater than about 0.65, notgreater than about 0.6, not greater than about 0.55, not greater thanabout 0.5, not greater than about 0.45, not greater than about 0.4, notgreat not greater than about 0.35, not greater than about 0.3, notgreater than about 0.35, not greater than about 0.3, not greater thanabout 0.25, not greater than about 0.2, not greater than about 0.15, notgreater than about 0.1, or even not greater than about 0.05. For yetanother non-limiting embodiment, the length factor (Los3/Lss3) can be atleast about 0.05, such as at least about 0.1, at least about 0.15, atleast about 0.2, at least about 0.25, at least about 0.3, at least about0.35, at least about 0.4, at least about 0.45, at least about 0.5, atleast about 0.55, at least about 0.6, at least about 0.65, at leastabout 0.7, at least about 0.75, at least about 0.8, at least about 0.85,at least about 0.9, or even at least about 0.95. It will be appreciatedthat the length factor (Los3/Lss3) can be within a range between any ofthe minimum and maximum values noted above.

In an alternative embodiment, the relationship between the length of thethird oblique side section (Los3) and the length of the third sidesection (Lss3) can define a length factor (Lss2/Los2) that mayfacilitate improved performance of the shaped abrasive particle 1300.For example, the length factor (Lss3/Los3) can be not greater than about1, such as not greater than about 0.95, not greater than about 0.9, notgreater than about 0.85, not greater than about 0.8, not greater thanabout 0.75, not greater than about 0.7, not greater than about 0.65, notgreater than about 0.6, not greater than about 0.55, not greater thanabout 0.5, not greater than about 0.45, not greater than about 0.4, notgreat not greater than about 0.35, not greater than about 0.3, notgreater than about 0.35, not greater than about 0.3, not greater thanabout 0.25, not greater than about 0.2, not greater than about 0.15, notgreater than about 0.1, or even not greater than about 0.05. For yetanother non-limiting embodiment, the length factor (Lss3/Los3) can be atleast about 0.05, such as at least about 0.1, at least about 0.15, atleast about 0.2, at least about 0.25, at least about 0.3, at least about0.35, at least about 0.4, at least about 0.45, at least about 0.5, atleast about 0.55, at least about 0.6, at least about 0.65, at leastabout 0.7, at least about 0.75, at least about 0.8, at least about 0.85,at least about 0.9, or even at least about 0.95. It will be appreciatedthat the length factor (Lss3/Los3) can be within a range between any ofthe minimum and maximum values noted above.

Additionally, the length of the third side section (Lss3) relative tothe length of the first side section (Lss1) may be controlled tofacilitate improved performance of the shaped abrasive particle 1300. Inone embodiment, Lss3 can be different compared to Lss1. For example,Lss3 can be greater than Lss1. In still other embodiments, Lss3 can beless than Lss1. For yet another embodiment, such as illustrated in FIG.13A, Lss3 and Lss1 can be essentially the same compared to each other.

In another aspect, the length of the third side section (Lss3) relativeto the length of the second side section (Lss2) may be controlled tofacilitate improved performance of the shaped abrasive particle 1300. Inone embodiment, Lss3 can be different compared to Lss2. For example,Lss3 can be greater than Lss2. In still other embodiments, Lss3 can beless than Lss2. For yet another embodiment, such as illustrated in FIG.13A, Lss3 and Lss2 can be essentially the same compared to each other.

Moreover, the length of the third oblique side section (Los3) relativeto the length of the first oblique side section (Los1) may be controlledto facilitate improved performance of the shaped abrasive particle 1300.In one embodiment, Los3 can be different compared to Los1. For example,Los3 can be greater than Los1. In still other embodiments, Los3 can beless than Los1. For yet another embodiment, such as illustrated in FIG.13A, Los3 and Los1 can be essentially the same compared to each other.

For another embodiment, the length of the third oblique side section(Los3) relative to the length of the second oblique side section (Los2)may be controlled to facilitate improved performance of the shapedabrasive particle 1300. In one embodiment, Los3 can be differentcompared to Los2. For example, Los3 can be greater than Los2. In stillother embodiments, Los3 can be less than Los2. For yet anotherembodiment, such as illustrated in FIG. 13A, Los3 and Los2 can beessentially the same compared to each other.

As further illustrated, the first side section 1305 and the thirdoblique side section 1319 can be joined to each other at a thirdexternal corner 1321, which defines a third external corner angle Aec3.In certain instances, the third external corner angle Aec3 can bedifferent than a value of the third oblique angle Ao3. In at least oneembodiment, the third external corner angle Aec3 can be less than thevalue of the third oblique angle Ao3.

The third external corner angle Aec3 can be formed to have a particularvalue that may faciltiate improved performance of the shaped abrasiveparticle. For example, the third external corner angle Aec3 may be notgreater than about 130 degrees, such as not greater than about 125degrees, not greater than about 120 degrees, not greater than about 115degrees, not greater than about 110 degrees, not greater than about 105degrees, not greater than about 100 degrees, not greater than about 95degrees, not greater than about 94 degrees, or even not greater thanabout 93 degrees. Still, in at least one non-limiting embodiment, thethird external corner angle Aec3 can be at least about 50 degrees, suchas at least about 55 degrees, at least about 60 degrees, at least about65 degrees, at least about 70 degrees, at least about 80 degrees, oreven at least about 85 degrees. It will be appreciated that the thirdexternal corner angle Aec3 can have a value within a range between anyof the minimum and maximum values noted above. In one particularembodiment, the third external corner angle Aec3 can be substantiallyperpendicular.

The third external corner angle Aec3 and the third oblique angle Ao3 maybe formed to have a particular relationship with respect to each other,which may be described as a third angle factor (Aec3/Ao3) having aparticular value that may facilitate improved performance of the shapedabrasive particle 1300. For example, the third angle factor (Aec3/Ao3)may be not greater than about 1, such as not greater than about 0.95,not greater than about 0.9, not greater than about 0.85, not greaterthan about 0.8, not greater than about 0.75, not greater than about 0.7,not greater than about 0.65, not greater than about 0.6, not greaterthan about 0.55, not greater than about 0.5, not greater than about0.45, not greater than about 0.4, not great not greater than about 0.35,not greater than about 0.3, not greater than about 0.35, not greaterthan about 0.3, not greater than about 0.25, not greater than about 0.2,not greater than about 0.15, not greater than about 0.1, or even notgreater than about 0.05. In yet another embodiment, the third anglefactor (Aec3/Ao3) may be at least about 0.05, such as at least about0.1, at least about 0.15, at least about 0.2, at least about 0.25, atleast about 0.3, at least about 0.35, at least about 0.4, at least about0.45, at least about 0.5, at least about 0.55, at least about 0.6, atleast about 0.65, at least about 0.7, at least about 0.75, at leastabout 0.8, at least about 0.85, at least about 0.9, or even at leastabout 0.95. It will be appreciated that the third angle factor(Aec3/Ao3) may be within a range between any of the minimum and maximumvalues noted above.

FIG. 13B includes a top view of the shaped abrasive particle of FIG. 13Aaccording to an embodiment. The shaped abrasive particle 1300 can have abody 1301 having any of the features of the embodiments herein. Notably,the body 1301 has a Shape Index of approximately 0.63.

FIG. 13C includes a top view of a shaped abrasive particle according toan embodiment. The shaped abrasive particle 1350 can have a body 1351having the features of other shaped abrasive particles of embodimentsherein, including an upper major surface 1353 and a bottom major surface(not shown) opposite the upper major surface 1353. The upper majorsurface 1353 and the bottom major surface can be separated from eachother by at least one side surface 1354, which may include one or morediscrete side surface sections.

According to one embodiment, the body 1351 can be defined as anirregular hexagon, wherein the body has a hexagonal (i.e., six-sided)two dimensional shape as viewed in the plane of a length and a width ofthe body 1351, and wherein at least two of the side sections, such asside sections 1355 and 1356, have a different length with respect toeach other. Moreover, as illustrated, none of the sides are parallel toeach other. And furthermore, while not illustrated, any of the sides mayhave a curvature to them, including a concave curvature wherein thesides may curve inwards toward the midpoint of the body 1351 betweencorners joining two sides.

The body 1351 can have an oblique, truncated shape as viewed top-down,and more particularly, can have an oblique, truncated shape with atleast one portion of the side surface 1354 that is curved. The body 1351can have any of the features of the body 1300 of the shaped abrasiveparticle of FIG. 13A. In one embodiment, the side surface 1354 caninclude a first side section 1355 and a first oblique side section 1356,which can be joined to each other at a first oblique corner 1357defining a first oblique corner angle Ao1, which may have an obtusevalue. Notably, the first side section 1355 can have a substantiallylinear contour. The first oblique side section 1356 can be substantiallynon-linear, such that at least a portion of the first oblique sidesection comprises a curvature. In one embodiment, the entire length ofthe first oblique side section 1356 can have a curvature. For example,the entire length of the first oblique side section 1356 extendingbetween the first oblique corner 1357 and the first exterior corner 1359can be curved. In a more particular embodiment, the first oblique sidesection 1356 can have a curvature, and that curvature can define amonotonic curve. The first oblique side section 1356 may define aconcave curvature, such that the portion of the body defined by thefirst oblique side section 1356 extends inward toward a midpoint 1381 ofthe body 1351.

In another instance, the first oblique side section 1356 can have acurvature defining an arc segment of a circle and defining a radius ofthe first oblique side section (Ros1). The size of the radius (Ros1) ofthe first oblique side section 1356 may be controlled to facilitateimproved performance of the body 1351. According to at least oneembodiment, the radius of the first oblique side section (Ros1) can bedifferent than the length of the first oblique side section (Los1),wherein Los1 is measured as the shortest linear distance between thecorners 1357 and 1359. In more particular instances, the radius of thefirst oblique side section (Ros1) can be greater than the length of thefirst oblique side section (Los1). The relationship between Ros1 andLos1 can be the same as the relationship between Lss1 and Los1 asdefined in the embodiments herein.

In yet another embodiment, the radius of the first oblique side section(Ros1) can be controlled relative to the length of the first sidesection (Lss1), which may facilitate improved performance of the body1351. For example, the radius of the first oblique side section (Ros1)can be different than the length of the first side section (Lss1). Inparticular, the relationship between Ros1 and Lss1 can be the same asthe relationship between Lss1 and Los1 as defined in the embodimentsherein. In particular instances, the radius of the first oblique sidesection (Ros1) can be greater than the length of the first side section(Lss1). Still, in another embodiment, the radius of the first obliqueside section (Ros1) can be less than the length of the first sidesection (Lss1).

In still another aspect, the radius of the first oblique side section(Ros1) can be controlled relative to the total length of the first side,including the length of the first side section (Lss1) and the length ofthe first oblique side section (Los1), which may facilitate improvedperformance of the body 1351. For example, the radius of the firstoblique side section (Ros1) can be different than the total length ofthe first side section (Lss1) and the first oblique side section (Los1).In particular instances, the radius of the first oblique side section(Ros1) can be greater than the total length of the first side section(Lss1) and the first oblique side section (Los1). Still, in anotherembodiment, the radius of the first oblique side section (Ros1) can beless than the total length of the first side section (Lss1) and thefirst oblique side section (Los1).

According to one embodiment, the radius of the first oblique sidesection can be not greater than 10 mm, such as not greater than 9 mm ornot greater than 8 mm or not greater than 7 mm or not greater than 6 mmor not greater than 5 mm or not greater than 4 mm or not greater than 3mm or even not greater than 2 mm. Still, in at least one non-limitingembodiment, the radius of the first oblique side section (Ros1) can beat least 0.01 mm, such as at least 0.05 mm or at least 0.1 mm or atleast 0.5 mm. It will be appreciated that the radius of the firstoblique side section can be within a range including any of the minimumand maximum values noted above.

Any reference to the angles of the body, including for example the firstoblique angle (Ao1), first external corner angle (Aec1), second obliqueangle (Ao2), second external corner angle (Aec2), third oblique angle(Ao3), and third external corner angle (Aec3) can be the same asprovided in the embodiments herein. Notably, provision of at least oneoblique side section having a curvature can reduce the angle at theadjoining corners where the curved section terminates (e.g., corners1357 and 1359). As illustrated, the angle of the first external corner(Aec1) can be measured as the angle created by the second side section1361 and the tangent 1358 to the first oblique side section 1356 at thecorner 1359 which is shown by the dotted line. Moreover, the provisionof a first oblique side section 1356 having a curvature can facilitate alower rake angle and improved grinding performance at the corner 1359for the body 1351 in the orientation as shown or in the mirror image ofthe orientation of the body 1351 as shown in FIG. 13C. Reduction in therake angle for multiple orientations may faciltiate improved grindingperformance by the body 1351 in a variety of orientations.

As further illustrated, the body 1351 can include a second side section1361 and second oblique side section 1362 joined to each other at thecorner 1363, which may define a second oblique corner angle (Ao2), whichmay have an obtuse value. The second side section 1361 can be coupled tothe first oblique side section 1356 at the first external corner 1359,wherein the first external corner 1359 defines the first external cornerangle (Aec1) and wherein the first external corner angle (Aec1) isdifferent than a value of the first oblique angle (Ao1) as described inaccordance with other embodiments herein. The first external corner 1359can be defined by a joint between a curved portion of the first obliqueside section 1356 and a linear portion of the second side section 1362.

As further illustrated, and according to an embodiment, at least aportion of the second oblique side section 1362 comprises a curvature,and more particularly, the entire length of the second oblique sidesection 1362 can have a curvature. In at least one embodiment, thesecond oblique side section 1362 can have a monotonic curve. The secondoblique side section 1362 can have a curvature defining an arc segmentof a circle and defining a radius of the second oblique side section(Ros2). In at least one embodiment, Ros1 and Ros2 can be substantiallythe same. Moreover, the relative curvature of the first oblique sidesection 1356 can be substantially the same as the curvature of thesecond oblique side section 1362. Still, in another embodiment, Ros1 andRos2 can be different compared to each other. Moreover, the relativecurvature of the first oblique side section 1356 can be differentcompared to the curvature of the second oblique side section 1362.

The body 1351 can include a third side section 1371 and third obliqueside section 1372 joined to each other at the corner 1373, which maydefine a third oblique corner angle angle (Ao3), which may have anobtuse value. The third side section 1371 can be coupled to the secondoblique side section 1362 at the second external corner 1364, whereinthe second external corner 1364 defines the second external corner angle(Aec2), which can have any of the attributes of similar corners ofshaped abrasive particles described herein. The second external corner1364 can be defined by a joint between a curved portion of the secondoblique side section 1362 and a linear portion of the third side section1372. The body also includes a third external corner 1374 between thethird oblique side section 1372 and the first side section 1355. Thethird external corner 1374 can define a third external corner angle(Aec3), which can have any of the attributes of similar cornersdescribed in embodiments herein. Moreover, the third side section 1371,third oblique side section 1372, and radius of the third oblique sidesection can have any of the same features of corresponding elementsdescribed in the embodiments herein

In yet another embodiment, the body 1301 can have at least one centralaxis 1382 extending from an external corner (e.g., corner 1364) andthrough the midpoint 1381 of the body 1351 to bisect the body 1351.According to one embodiment, the body 1351 can be asymmetric about thecentral axis 1382. That is, the shape of the body 1351 as defined by thecontour of the side surface 1354 as viewed top down on either side ofthe central axis 1382 are not identical, and therefore, the central axis1382 defines an axis of asymmetry. In other instances, the body can havemore than one central axis defining an axis of asymmetry, including forexample, at least three different central axes, wherein the body isasymmetric about each of the three different central axes.

The shaped abrasive particles of the embodiments herein, including butnot limited to the body 1351 of the shaped abrasive particle 1350 canhave a side surface including at least 5 different side sections,wherein the 5 different side sections are separated by a corner, whichmay be an external corner. External corners are those corners over whicha hypothetical rubber band would be deflected. That is, if ahypothetical rubber band were placed around the side surface 1354 of thebody 1351, it should be deflected around the corners 1357, 1359, 1363,1364, 1373, and 1374. Each of the external corners 1357, 1359, 1363,1364, 1373, and 1374 define and separate distinct side sections of theside surface 1354. In at least one embodiment, the side surface 1354 ofthe body 1351 comprises at least two linear portions separated by atleast one curved portion. For example, the body 1351 can include a firstside section 1355 and a second side section 1361 separated from eachother by the first oblique side section 1356. In still anotherembodiment, the side surface 1354 of the body 1351 comprises linearportions and curved portions which are alternating with respect to eachother. For example, the side surface 1354 of the body 1351 compriseslinear portions and curved portions and wherein each linear portion isjoined to at least one curved portion, and furthermore, may be connectedto each other at an exterior corner. The body 1351 does not necessarilyhave two linear portions directly connected to each other or two curvedportions directly connected to each other. It will be appreciated thatthis is true for one non-limiting embodiment, and other shapes may havelinear portions and/or curved portions directly connected to each other.

In a particular instance, the shaped abrasive particles of theembodiments herein can have a particular draft angle at the intersectionof the smallest major surface and the side surface, which may beindicative of a particular aspect of forming and/or may facilitateimproved performance of the abrasive particle. In one particularinstance, the shaped abrasive particles herein can have an average draftangle, which can be an average measure of draft angle for astatistically relevant and random sample size of shaped abrasiveparticles (e.g., at least 20 particles). In a particular instance, theaverage draft angle can be not greater than 95°, such as not greaterthan 94° or no greater than 93° or not greater than 92° or not greaterthan 91° or even not greater than 90°. In at least one non-limitingembodiment, the shaped abrasive particles of the embodiments herein canhave an average draft angle of at least 80° such as at least 82° or atleast 84° or at least 85° or at least 86° or at least 87°. It will beappreciated that the shaped abrasive particles of the embodiments hereincan have an average draft angle within a range including any of theminimum and maximum values noted above, including but not limited to,within a range of at least 80° and not greater than 95° or within arange including at least 80° and not greater than 94° or within a rangeincluding at least 82° and not greater than 93° or within a rangeincluding at least 84° and not greater than 93°.

The draft angle can be measured by cutting the shaped abrasive particlein half at an approximately 90° angle with respect to the major surfaceand at a perpendicular angle to one of the side surfaces, such as shownby the dotted line in FIG. 13D. As best as possible, the sectioning lineshould extend perpendicular to the side surface and through the midpointof a major surface of the particle. The portion of the shaped abrasiveparticle is then mounted and viewed via SEM in a manner that is similarto that provided in FIG. 13E. A suitable program for such includesImageJ software. Using the image of the body, the smallest major surfaceis determined by identifying the largest major surface and selecting thesurface opposite thereof. Certain shaped abrasive particles may have agenerally square cross-sectional shape. To identify the smallest majorsurface, the largest major surface must first be determined.

The smallest major surface is that surface opposite the largest majorsurface. The imaging software, such as ImageJ may be utilized to assistwith the determination of the smallest major surface. Using a suitableimage processing software (e.g., ImageJ) draw a straight line along bothof the major surfaces between the corners adjoining the major surfacesand the sidewall as provided by the lines below in FIG. 13E. Using theimage analysis software, measure the line that longer. The shorter ofthe two lines is presumed to be the smaller of the two major surfaces.In the case provided in FIG. 13E, the line on the right of the image isshorter and the draft angle should be measured at the corner identifiedat the upper right-hand corner, which is also illustrated in FIG. 13F.

To measure the draft angle, lines can be drawn along the smallest majorsurface and the side surface to form an intersecting angle as providedin FIG. 13F. The lines are drawn taking into consideration the shape ofthe surfaces as a whole and ignoring imperfections or othernon-representative surface undulations at the corner of the particle(e.g., cracks or chips due to mounting procedures, etc.). Moreover, theline representing the smaller major surface is drawn to represent theportion of the major surface that connects the sidewall at the draftangle. The draft angle (i.e., the angle of the body as measured at theintersection) is determined by the interior angle formed at theintersection of the lines.

FIG. 14 includes a top view of another shaped abrasive particleaccording to an embodiment. The shaped abrasive particle 1400 can have abody 1401 having any of the features of the embodiments herein. Notably,the body 1401 has a regular heptagonal (i.e., seven-sided) twodimensional shape as viewed in the plane of a length and a width of thebody 1401, wherein all of the sides are substantially the same lengthwith respect to each other. The body 1401 has a Shape Index ofapproximately 0.9.

FIG. 15 through FIG. 26 include other shaped abrasive particlesaccording to the embodiments herein. In particular, the shaped abrasiveparticle of FIG. 15 has a Shape Index of 0.66. The shaped abrasiveparticles of FIGS. 16, 17 and 18 have a Shape Index of approximately0.81. In particular, the shaped abrasive particle of FIG. 19 has a ShapeIndex of 0.57. The shaped abrasive particle of FIG. 20 has a Shape Indexof approximately 0.69. The shaped abrasive particle of FIG. 21 has aShape Index of approximately 0.77. The shaped abrasive particle of FIG.22 has a Shape Index of approximately 0.62. The shaped abrasive particleof FIG. 23 has a Shape Index of approximately 0.57. The shaped abrasiveparticle of FIG. 24 has a Shape Index of approximately 0.67. The shapedabrasive particle of FIG. 25 has a Shape Index of approximately 0.82.The shaped abrasive particle of FIG. 26 has a Shape Index ofapproximately 0.77.

A Fixed Abrasive Article

After forming or sourcing the shaped abrasive particles, the particlescan be combined with other materials to form a fixed abrasive article.In a fixed abrasive, the shaped abrasive particles can be coupled to amatrix or substrate and used for material removal operations. Somesuitable exemplary fixed abrasive articles can include bonded abrasivearticles wherein the shaped abrasive particles are contained in a threedimensional matrix of bond material. In other instances, the fixedabrasive article may be a coated abrasive article, wherein the shapedabrasive particles may be dispersed in a single layer overlying abacking and bonded to the backing using one or more adhesive layers.

FIG. 5A includes an illustration of a bonded abrasive articleincorporating the abrasive particulate material in accordance with anembodiment. As illustrated, the bonded abrasive 590 can include a bondmaterial 591, abrasive particulate material 592 contained in the bondmaterial, and porosity 598 within the bond material 591. In particularinstances, the bond material 591 can include an organic material,inorganic material, and a combination thereof. Suitable organicmaterials can include polymers, such as epoxies, resins, thermosets,thermoplastics, polyimides, polyamides, and a combination thereof.Certain suitable inorganic materials can include metals, metal alloys,vitreous phase materials, crystalline phase materials, ceramics, and acombination thereof.

In some instances, the abrasive particulate material 592 of the bondedabrasive 590 can include shaped abrasive particles 593, 594, 595, and596. In particular instances, the shaped abrasive particles 593, 594,595, and 596 can be different types of particles, which can differ fromeach other in composition, two-dimensional shape, three-dimensionalshape, size, and a combination thereof as described in the embodimentsherein. Alternatively, the bonded abrasive article can include a singletype of shaped abrasive particle.

The bonded abrasive 590 can include a type of abrasive particulatematerial 597 representing diluent abrasive particles, which can differfrom the shaped abrasive particles 593, 594, 595, and 596 incomposition, two-dimensional shape, three-dimensional shape, size, and acombination thereof.

The porosity 598 of the bonded abrasive 590 can be open porosity, closedporosity, and a combination thereof. The porosity 598 may be present ina majority amount (vol %) based on the total volume of the body of thebonded abrasive 590. Alternatively, the porosity 598 can be present in aminor amount (vol %) based on the total volume of the body of the bondedabrasive 590. The bond material 591 may be present in a majority amount(vol %) based on the total volume of the body of the bonded abrasive590. Alternatively, the bond material 591 can be present in a minoramount (vol %) based on the total volume of the body of the bondedabrasive 590. Additionally, abrasive particulate material 592 can bepresent in a majority amount (vol %) based on the total volume of thebody of the bonded abrasive 590. Alternatively, the abrasive particulatematerial 592 can be present in a minor amount (vol %) based on the totalvolume of the body of the bonded abrasive 590.

FIG. 5B includes a cross-sectional illustration of a coated abrasivearticle in accordance with an embodiment. In particular, the coatedabrasive article 500 can include a substrate 501 (e.g., a backing) andat least one adhesive layer overlying a surface of the substrate 501.The adhesive layer can include a make coat 503 and/or a size coat 504.The coated abrasive article 500 can include abrasive particulatematerial 510, which can include shaped abrasive particles 505 of any ofthe embodiments herein and a second type of abrasive particulatematerial 507 in the form of diluent abrasive particles having a randomshape, which may not necessarily be shaped abrasive particles. Theshaped abrasive particles 505 of FIG. 5B are illustrated generally forpurposes or discussion, and it will be appreciated that the coatedabrasive article can include any shaped abrasive particles of theembodiments herein. The make coat 503 can be overlying the surface ofthe substrate 501 and surrounding at least a portion of the shapedabrasive particles 505 and second type of abrasive particulate material507. The size coat 504 can be overlying and bonded to the shapedabrasive particles 505 and second type of abrasive particulate material507 and the make coat 503.

According to one embodiment, the substrate 501 can include an organicmaterial, inorganic material, and a combination thereof. In certaininstances, the substrate 501 can include a woven material. However, thesubstrate 501 may be made of a non-woven material. Particularly suitablesubstrate materials can include organic materials, including polymerssuch as polyester, polyurethane, polypropylene, and/or polyimides suchas KAPTON from DuPont, and paper. Some suitable inorganic materials caninclude metals, metal alloys, and particularly, foils of copper,aluminum, steel, and a combination thereof. The backing can include oneor more additives selected from the group of catalysts, coupling agents,currants, anti-static agents, suspending agents, anti-loading agents,lubricants, wetting agents, dyes, fillers, viscosity modifiers,dispersants, defoamers, and grinding agents.

A polymer formulation may be used to form any of a variety of layers ofthe coated abrasive article 500 such as, for example, a frontfill, apre-size, the make coat, the size coat, and/or a supersize coat. Whenused to form the frontfill, the polymer formulation generally includes apolymer resin, fibrillated fibers (preferably in the form of pulp),filler material, and other optional additives. Suitable formulations forsome frontfill embodiments can include material such as a phenolicresin, wollastonite filler, defoamer, surfactant, a fibrillated fiber,and a balance of water. Suitable polymeric resin materials includecurable resins selected from thermally curable resins including phenolicresins, urea/formaldehyde resins, phenolic/latex resins, as well ascombinations of such resins. Other suitable polymeric resin materialsmay also include radiation curable resins, such as those resins curableusing electron beam, UV radiation, or visible light, such as epoxyresins, acrylated oligomers of acrylated epoxy resins, polyester resins,acrylated urethanes and polyester acrylates and acrylated monomersincluding monoacrylated, multiacrylated monomers. The formulation canalso comprise a nonreactive thermoplastic resin binder which can enhancethe self-sharpening characteristics of the deposited abrasive particlesby enhancing the erodability. Examples of such thermoplastic resininclude polypropylene glycol, polyethylene glycol, andpolyoxypropylene-polyoxyethene block copolymer, etc. Use of a frontfillon the substrate 501 can improve the uniformity of the surface, forsuitable application of the make coat 503 and improved application andorientation of shaped abrasive particles 505 in a predeterminedorientation.

The make coat 503 can be applied to the surface of the substrate 501 ina single process, or alternatively, the abrasive particulate material510 can be combined with a make coat 503 material and applied as amixture to the surface of the substrate 501. Suitable materials of themake coat 503 can include organic materials, particularly polymericmaterials, including for example, polyesters, epoxy resins,polyurethanes, polyamides, polyacrylates, polymethacrylates, polyvinylchlorides, polyethylene, polysiloxane, silicones, cellulose acetates,nitrocellulose, natural rubber, starch, shellac, and mixtures thereof.In one embodiment, the make coat 503 can include a polyester resin. Thecoated substrate can then be heated in order to cure the resin and theabrasive particulate material to the substrate. In general, the coatedsubstrate 501 can be heated to a temperature of between about 100° C. toless than about 250° C. during this curing process.

The abrasive particulate material 510 can include shaped abrasiveparticles 505 according to embodiments herein. In particular instances,the abrasive particulate material 510 may include different types ofshaped abrasive particles 505. The different types of shaped abrasiveparticles can differ from each other in composition, in two-dimensionalshape, in three-dimensional shape, in size, and a combination thereof asdescribed in the embodiments herein. As illustrated, the coated abrasive500 can include a shaped abrasive particle 505, which may have any ofthe shapes of the shaped abrasive particles of the embodiments herein.

The other type of abrasive particles 507 can be diluent particlesdifferent than the shaped abrasive particles 505. For example, thediluent particles can differ from the shaped abrasive particles 505 incomposition, in two-dimensional shape, in three-dimensional shape, insize, and a combination thereof. For example, the abrasive particles 507can represent conventional, crushed abrasive grit having random shapes.The abrasive particles 507 may have a median particle size less than themedian particle size of the shaped abrasive particles 505.

After sufficiently forming the make coat 503 with the abrasiveparticulate material 510, the size coat 504 can be formed to overlie andbond the abrasive particulate material 510 in place. The size coat 504can include an organic material, may be made essentially of a polymericmaterial, and notably, can use polyesters, epoxy resins, polyurethanes,polyamides, polyacrylates, polymethacrylates, poly vinyl chlorides,polyethylene, polysiloxane, silicones, cellulose acetates,nitrocellulose, natural rubber, starch, shellac, and mixtures thereof.

According to one embodiment, the shaped abrasive particles 505 can beoriented in a predetermined orientation relative to each other and/orthe substrate 501. While not completely understood, it is thought thatone or a combination of dimensional features may be responsible forimproved orientation of the shaped abrasive particles 505. According toone embodiment, the shaped abrasive particles 505 can be oriented in aflat orientation relative to the substrate 501, such as that shown inFIG. 5B. In the flat orientation, the bottom surface 304 of the shapedabrasive particles can be closest to a surface of the substrate 501 andthe upper surface 303 of the shaped abrasive particles 505 can bedirected away from the substrate 501 and configured to conduct initialengagement with a workpiece.

According to another embodiment, the shaped abrasive particles 505 canbe placed on a substrate 501 in a predetermined side orientation, suchas that shown in FIG. 6. In particular instances, a majority of theshaped abrasive particles 505 of the total content of shaped abrasiveparticles 505 on the abrasive article 500 can have a predetermined sideorientation. In the side orientation, the bottom surface 304 of theshaped abrasive particles 505 can be spaced away from and angledrelative to the surface of the substrate 501. In particular instances,the bottom surface 304 can form an obtuse angle (B) relative to thesurface of the substrate 501. Moreover, the upper surface 303 is spacedaway and angled relative to the surface of the substrate 501, which inparticular instances, may define a generally acute angle (A). In a sideorientation, a side surface 305 can be closest to the surface of thesubstrate 501, and more particularly, may be in direct contact with asurface of the substrate 501.

For certain other abrasive articles herein, at least about 55% of theplurality of shaped abrasive particles 505 on the abrasive article 500can be coupled to the backing in a predetermined side orientation.Still, the percentage may be greater, such as at least about 60%, atleast about 65%, at least about 70%, at least about 75%, at least about77%, at least about 80%, at least about 81%, or even at least about 82%.And for one non-limiting embodiment, an abrasive article 500 may beformed using the shaped abrasive particles 505 herein, wherein notgreater than about 99% of the total content of shaped abrasive particleshave a predetermined side orientation.

To determine the percentage of particles in a predetermined orientation,a 2D microfocus x-ray image of the abrasive article 500 is obtainedusing a CT scan machine run in the conditions of Table 1 below. TheX-ray 2D imaging is conducted on shaped abrasive particles on a backingwith Quality Assurance software. A specimen mounting fixture utilizes aplastic frame with a 4″×4″ window and an Ø0.5″ solid metallic rod, thetop part of which is half flattened with two screws to fix the frame.Prior to imaging, a specimen is clipped over one side of the frame wherethe screw heads face the incidence direction of the X-rays. Then fiveregions within the 4″×4″ window area are selected for imaging at 120kV/80 μA. Each 2D projection is recorded with the X-ray off-set/gaincorrections and at a magnification of 15 times.

TABLE 1 Field of view Voltage Current per image Exposure (kV) (μA)Magnification (mm × mm) time 120 80 15X 16.2 × 13.0 500 ms/2.0 fps

The image is then imported and analyzed using the ImageJ program,wherein different orientations are assigned values according to Table 2below. FIG. 11 includes images representative of portions of a coatedabrasive article according to an embodiment, which images can be used toanalyze the orientation of shaped abrasive particles on the backing.

TABLE 2 Cell marker type Comments 1 Grains on the perimeter of theimage, partially exposed - standing up 2 Grains on the perimeter of theimage, partially exposed - down 3 Grains on the image, completelyexposed - standing vertical 4 Grains on the image, completely exposed -down 5 Grains on the image, completely exposed - standing slanted(between standing vertical and down)

Three calculations are then performed as provided below in Table 3.After conducting the calculations, the percentage of grains in aparticular orientation (e.g., side orientation) per square centimetercan be derived.

TABLE 3 5) Parameter Protocol* % grains up ((0.5 × 1) + 3 + 5)/(1 + 2 +3 + 4 + 5) Total # of grains per cm² (1 + 2 + 3 + 4 + 5) # of grains upper cm² (% grains up × Total # of grains per cm² *These are allnormalized with respect to the representative area of the image. + - Ascale factor of 0.5 was applied to account for the fact that they arenot completely present in the image.

Furthermore, the abrasive articles made with the shaped abrasiveparticles can utilize various contents of the shaped abrasive particles.For example, the abrasive articles can be coated abrasive articlesincluding a single layer of a plurality of shaped abrasive particles inan open-coat configuration or a closed-coat configuration. For example,the plurality of shaped abrasive particles can define an open-coatabrasive article having a coating density of shaped abrasive particlesof not greater than about 70 particles/cm². In other instances, theopen-coat density of shaped abrasive particles per square centimeter ofabrasive article may be not greater than about 65 particles/cm², such asnot greater than about 60 particles/cm², not greater than about 55particles/cm², or even not greater than about 50 particles/cm². Still,in one non-limiting embodiment, the density of the open-coat abrasivearticle using the shaped abrasive particle herein can be at least about5 particles/cm², or even at least about 10 particles/cm². It will beappreciated that the open-coat density of the coated abrasive articlecan be within a range between any of the above minimum and maximumvalues.

In an alternative embodiment, the plurality of shaped abrasive particlescan define a closed-coat abrasive article having a coating density ofshaped abrasive particles of at least about 75 particles/cm², such as atleast about 80 particles/cm², at least about 85 particles/cm², at leastabout 90 particles/cm², at least about 100 particles/cm². Still, in onenon-limiting embodiment, the closed-coat density of the coated abrasivearticle using the shaped abrasive particle herein can be not greaterthan about 500 particles/cm². It will be appreciated that the closedcoat density of the coated abrasive article can be within a rangebetween any of the above minimum and maximum values.

In certain instances, the abrasive article can have an open-coat densityof a coating not greater than about 50% of abrasive particulate materialcovering the exterior abrasive surface of the article. In otherembodiments, the percentage coating of the abrasive particulate materialrelative to the total area of the abrasive surface can be not greaterthan about 40%, not greater than about 30%, not greater than about 25%,or even not greater than about 20%. Still, in one non-limitingembodiment, the percentage coating of the abrasive particulate materialrelative to the total area of the abrasive surface can be at least about5%, such as at least about 10%, at least about 15%, at least about 20%,at least about 25%, at least about 30%, at least about 35%, or even atleast about 40%. It will be appreciated that the percent coverage ofshaped abrasive particles for the total area of abrasive surface can bewithin a range between any of the above minimum and maximum values.

Some abrasive articles may have a particular content of abrasiveparticles for a length (e.g., ream) of the backing or the substrate 501.For example, in one embodiment, the abrasive article may utilize anormalized weight of shaped abrasive particles of at least about 20lbs/ream, such as at least about 25 lbs/ream, or even at least about 30lbs/ream. Still, in one non-limiting embodiment, the abrasive articlescan include a normalized weight of shaped abrasive particles of notgreater than about 60 lbs/ream, such as not greater than about 50lbs/ream, or even not greater than about 45 lbs/ream. It will beappreciated that the abrasive articles of the embodiments herein canutilize a normalized weight of shaped abrasive particles within a rangebetween any of the above minimum and maximum values.

The plurality of shaped abrasive particles on an abrasive article asdescribed herein can define a first portion of a batch of abrasiveparticles, and the features described in the embodiments herein canrepresent features that are present in at least a first portion of abatch of shaped abrasive particles. Moreover, according to anembodiment, control of one or more process parameters as alreadydescribed herein also can control the prevalence of one or more featuresof the shaped abrasive particles of the embodiments herein. Theprovision of one or more features of any shaped abrasive particle of abatch may facilitate alternative or improved deployment of the particlesin an abrasive article and may further facilitate improved performanceor use of the abrasive article. The batch may also include a secondportion of abrasive particles. The second portion of abrasive particlescan include diluent particles.

In accordance with one aspect of the embodiments herein, a fixedabrasive article can include a blend of abrasive particles. The blend ofabrasive particles can include a first type of shaped abrasive particleand a second type of shaped abrasive particle. The first type of shapedabrasive particle can include any features of the shaped abrasiveparticles of the embodiments herein. The second type of shaped abrasiveparticle can include any features of the shaped abrasive particles ofthe embodiments herein. Moreover, it will be appreciated in light of thepresent disclosure that one or more different types of abrasiveparticles, including abrasive particles of the embodiments herein and/orconventional abrasive particles may be combined in a fixed abrasive toimprove the overall performance of the abrasive article. This mayinclude the use of blends of different types of abrasive particles,wherein the different types of abrasive particles may differ in size,shape, hardness, fracture toughness, strength, tip sharpness, ShapeIndex, composition, type and/or content of dopants, and a combinationthereof.

The blend of abrasive particles can include a first type of shapedabrasive particle present in a first content (C1), which may beexpressed as a percentage (e.g., a weight percent) of the first type ofshaped abrasive particles as compared to the total content of particlesof the blend. Furthermore, the blend of abrasive particles may include asecond content (C2) of the second type of shaped abrasive particles,expressed as a percentage (e.g., a weight percent) of the second type ofshaped abrasive particles relative to the total weight of the blend. Thefirst content can be the same as or different from the second content.For example, in certain instances, the blend can be formed such that thefirst content (C1) can be not greater than about 90% of the totalcontent of the blend. In another embodiment, the first content may beless, such as not greater than about 85%, not greater than about 80%,not greater than about 75%, not greater than about 70%, not greater thanabout 65%, not greater than about 60%, not greater than about 55%, notgreater than about 50%, not greater than about 45%, not greater thanabout 40%, not greater than about 35%, not greater than about 30%, notgreater than about 25%, not greater than about 20%, not greater thanabout 15%, not greater than about 10%, or even not greater than about5%. Still, in one non-limiting embodiment, the first content of thefirst type of shaped abrasive particles may be present in at least about1% of the total content of abrasive particles of the blend. In yet otherinstances, the first content (C1) may be at least about 5%, such as atleast about 10%, at least about 15%, at least about 20%, at least about25%, at least about 30%, at least about 35%, at least about 40%, atleast about 45%, at least about 50%, at least about 55%, at least about60%, at least about 65%, at least about 70%, at least about 75%, atleast about 80%, at least about 85%, at least about 90%, or even atleast about 95%. It will be appreciated that the first content (C1) maybe present within a range between any of the minimum and maximumpercentages noted above.

The blend of abrasive particles may include a particular content of thesecond type of shaped abrasive particle. For example, the second content(C2) may be not greater than about 98% of the total content of theblend. In other embodiments, the second content may be not greater thanabout 95%, such as not greater than about 90%, not greater than about85%, not greater than about 80%, not greater than about 75%, not greaterthan about 70%, not greater than about 65%, not greater than about 60%,not greater than about 55%, not greater than about 50%, not greater thanabout 45%, not greater than about 40%, not greater than about 35%, notgreater than about 30%, not greater than about 25%, not greater thanabout 20%, not greater than about 15%, not greater than about 10%, oreven not greater than about 5%. Still, in one non-limiting embodiment,the second content (C2) may be present in an amount of at least about 1%of the total content of the blend. For example, the second content maybe at least about 5%, such as at least about 10%, at least about 15%, atleast about 20%, at least about 25%, at least about 30%, at least about35%, at least about 40%, at least about 45%, at least about 50%, atleast about 55%, at least about 60%, at least about 65%, at least about70%, at least about 75%, at least about 80%, at least about 85%, atleast about 90%, or even at least about 95%. It will be appreciated thatthe second content (C2) can be within a range between any of the minimumand maximum percentages noted above.

In accordance with another embodiment, the blend of abrasive particlesmay have a blend ratio (C1/C2) that may define a ratio between the firstcontent (C1) and the second content (C2). For example, in oneembodiment, the blend ratio (C1/C2) may be not greater than about 10. Inyet another embodiment, the blend ratio (C1/C2) may be not greater thanabout 8, such as not greater than about 6, not greater than about 5, notgreater than about 4, not greater than about 3, not greater than about2, not greater than about 1.8, not greater than about 1.5, not greaterthan about 1.2, not greater than about 1, not greater than about 0.9,not greater than about 0.8, not greater than about 0.7, not greater thanabout 0.6, not greater than about 0.5, not greater than about 0.4, notgreater than about 0.3, or even not greater than about 0.2. Still, inanother non-limiting embodiment, the blend ratio (C1/C2) may be at leastabout 0.1, such as at least about 0.15, at least about 0.2, at leastabout 0.22, at least about 0.25, at least about 0.28, at least about0.3, at least about 0.32, at least about 0.3, at least about 0.4, atleast about 0.45, at least about 0.5, at least about 0.55, at leastabout 0.6, at least about 0.65, at least about 0.7, at least about 0.75,at least about 0.8, at least about 0.9, at least about 0.95, at leastabout 1, at least about 1.5, at least about 2, at least about 3, atleast about 4, or even at least about 5. It will be appreciated that theblend ratio (C1/C2) may be within a range between any of the minimum andmaximum values noted above.

In at least one embodiment, the blend of abrasive particles can includea majority content of shaped abrasive particles. That is, the blend canbe formed primarily of shaped abrasive particles, including, but notlimited to, a first type of shaped abrasive particle and a second typeof shaped abrasive particle. In at least one particular embodiment, theblend of abrasive particles can consist essentially of the first type ofshaped abrasive particle and the second type of shaped abrasiveparticle. However, in other non-limiting embodiments, the blend mayinclude other types of abrasive particles. For example, the blend mayinclude a third type of abrasive particle that may include aconventional abrasive particle or a shaped abrasive particle. The thirdtype of abrasive particle may include a diluent type of abrasiveparticle having an irregular shape, which may be achieved throughconventional crushing and comminution techniques.

According to another embodiment, the blend of abrasive particles caninclude a plurality of shaped abrasive particles and each of the shapedabrasive particles of the plurality may be arranged in a controlledorientation relative to a backing, such as a substrate of a coatedabrasive article. Suitable exemplary controlled orientations can includeat least one of a predetermined rotational orientation, a predeterminedlateral orientation, and a predetermined longitudinal orientation. In atleast one embodiment, the plurality of shaped abrasive particles havinga controlled orientation can include at least a portion of the firsttype of shaped abrasive particles of the blend, at least a portion ofthe second type of shaped abrasive particles of the blend, and acombination thereof. More particularly, the plurality of shaped abrasiveparticles having a controlled orientation can include all of the firsttype of shaped abrasive particles. In still another embodiment, theplurality of shaped abrasive particles arranged in a controlledorientation relative to the backing may include all of the second typeof shaped abrasive particles within the blend of abrasive particles.

FIG. 7 includes a top view illustration of a portion of a coatedabrasive article including shaped abrasive particles having controlledorientation. As illustrated, the coated abrasive article 700 includes abacking 701 that can be defined by a longitudinal axis 780 that extendsalong and defines a length of the backing 701 and a lateral axis 781that extends along and defines a width of the backing 701. In accordancewith an embodiment, a shaped abrasive particle 702 can be located in afirst, predetermined position 712 defined by a particular first lateralposition relative to the lateral axis of 781 of the backing 701 and afirst longitudinal position relative to the longitudinal axis 780 of thebacking 701. Furthermore, a shaped abrasive particle 703 may have asecond, predetermined position 713 defined by a second lateral positionrelative to the lateral axis 781 of the backing 701, and a firstlongitudinal position relative to the longitudinal axis 780 of thebacking 701 that is substantially the same as the first longitudinalposition of the shaped abrasive particle 702. Notably, the shapedabrasive particles 702 and 703 may be spaced apart from each other by alateral space 721, defined as a smallest distance between the twoadjacent shaped abrasive particles 702 and 703 as measured along alateral plane 784 parallel to the lateral axis 781 of the backing 701.In accordance with an embodiment, the lateral space 721 can be greaterthan zero, such that some distance exists between the shaped abrasiveparticles 702 and 703. However, while not illustrated, it will beappreciated that the lateral space 721 can be zero, allowing for contactand even overlap between portions of adjacent shaped abrasive particles.

As further illustrated, the coated abrasive article 700 can include ashaped abrasive particle 704 located at a third, predetermined position714 defined by a second longitudinal position relative to thelongitudinal axis 780 of the backing 701 and also defined by a thirdlateral position relative to a lateral plane 785 parallel to the lateralaxis 781 of the backing 701 and spaced apart from the lateral axis 784.Further, as illustrated, a longitudinal space 723 may exist between theshaped abrasive particles 702 and 704, which can be defined as asmallest distance between the two adjacent shaped abrasive particles 702and 704 as measured in a direction parallel to the longitudinal axis780. In accordance with an embodiment, the longitudinal space 723 can begreater than zero. Still, while not illustrated, it will be appreciatedthat the longitudinal space 723 can be zero, such that the adjacentshaped abrasive particles are touching, or even overlapping each other.

FIG. 8A includes a top view illustration of a portion of an abrasivearticle including shaped abrasive particles in accordance with anembodiment. As illustrated, the abrasive article 800 can include ashaped abrasive particle 802 overlying a backing 801 in a first positionhaving a first rotational orientation relative to a lateral axis 781defining the width of the backing 801. In particular, the shapedabrasive particle 802 can have a predetermined rotational orientationdefined by a first rotational angle between a lateral plane 884 parallelto the lateral axis 781 and a dimension of the shaped abrasive particle802. Notably, reference herein to a dimension of the shaped abrasiveparticle 802 can include reference to a bisecting axis 831 of the shapedabrasive particle 802, such bisecting axis 831 extending through acenter point 821 of the shaped abrasive particle 802 along a surface(e.g., a side or an edge) connected to (directly or indirectly) thebacking 801. Accordingly, in the context of a shaped abrasive particlepositioned in a side orientation, (see, e.g., FIG. 6), the bisectingaxis 831 can extend through a center point 821 and in the direction ofthe width (w) of a side 833 closest to the surface of the backing 801.

In certain embodiments, the predetermined rotational orientation of theshaped abrasive particle 802 can be defined by a predeterminedrotational angle 841 that defines the smallest angle between thebisecting axis 831 and the lateral plane 884, both of which extendthrough the center point 821 as viewed from the top down in FIG. 8A. Inaccordance with an embodiment, the predetermined rotational angle 841,and thus the predetermined rotational orientation, can be 0°. In otherembodiments, the predetermined rotational angle defining thepredetermined rotational orientation can be greater, such as at leastabout 2°, at least about 5°, at least about 10°, at least about 15°, atleast about 20°, at least about 25°, at least about 30°, at least about35°, at least about 40°, at least about 45°, at least about 50°, atleast about 55°, at least about 60°, at least about 70°, at least about80°, or even at least about 85°. Still, the predetermined rotationalorientation as defined by the rotational angle 841 may be not greaterthan about 90°, such as not greater than about 85°, not greater thanabout 80°, not greater than about 75°, not greater than about 70°, notgreater than about 65°, not greater than about 60°, such as not greaterthan about 55°, not greater than about 50°, not greater than about 45°,not greater than about 40°, not greater than about 35°, not greater thanabout 30°, not greater than about 25°, not greater than about 20°, suchas not greater than about 15°, not greater than about 10°, or even notgreater than about 5°. It will be appreciated that the predeterminedrotational orientation can be within a range between any of the aboveminimum and maximum angles.

FIG. 8B includes a perspective view illustration of a portion of theabrasive article 800 including the shaped abrasive particle 802 having atriangular two-dimensional shape. The referenced shaped abrasiveparticle having a triangular two-dimensional shape is merelyillustrative, and it will be appreciated that any shaped abrasiveparticle having any of the shapes of the embodiments herein can besubstituted for the triangular shaped abrasive particle of FIG. 8B. Asillustrated, the abrasive article 800 can include the shaped abrasiveparticle 802 overlying the backing 801 in a first position 812 such thatthe shaped abrasive particle 802 includes a first rotational orientationrelative to the lateral axis 781 defining the width of the backing 801.Certain aspects of the predetermined orientation of a shaped abrasiveparticle may be described by reference to a x, y, z three-dimensionalaxis as illustrated. For example, the predetermined longitudinalorientation of the shaped abrasive particle 802 may be described byreference to the position of the shaped abrasive particle 802 relativeto the y-axis, which extends parallel to the longitudinal axis 780 ofthe backing 801. Moreover, the predetermined lateral orientation of theshaped abrasive particle 802 may be described by reference to theposition of the shaped abrasive particle on the x-axis, which extendsparallel to the lateral axis 781 of the backing 801. Furthermore, thepredetermined rotational orientation of the shaped abrasive particle 802may be defined with reference to a bisecting axis 831 that extendsthrough the center point 821 of the side 833 of the shaped abrasiveparticle 802. Notably, the side 833 of the shaped abrasive particle 802may be connected either directly or indirectly to the backing 801. In aparticular embodiment, the bisecting axis 831 may form an angle with anysuitable reference axis including, for example, the x-axis that extendsparallel to the lateral axis 781. The predetermined rotationalorientation of the shaped abrasive particle 802 may be described as arotational angle formed between the x-axis and the bisecting axis 831,which rotational angle is depicted in FIG. 8B as angle 841. Notably, thecontrolled placement of a plurality of shaped abrasive particles on thebacking of the abrasive article may facilitate improved performance ofthe abrasive article.

FIG. 9 includes a perspective view illustration of a portion of anabrasive article including shaped abrasive particles havingpredetermined orientation characteristics relative to a grindingdirection in accordance with an embodiment. Notably, as with FIG. 8B,the shaped abrasive particles have a triangular two-dimensional shape,which is done merely for illustration and discussion of certain featuresof the abrasive article. It will be appreciated that any of shapedabrasive particles of the embodiments herein can be substituted for theshaped abrasive particles illustrated in FIG. 9. In one embodiment, theabrasive article 900 can include a shaped abrasive particle 902 having apredetermined orientation relative to another shaped abrasive particle903 and/or relative to a grinding direction 985. The grinding direction985 may be an intended direction of movement of the abrasive articlerelative to a workpiece in a material removal operation. In particularinstances, the grinding direction 985 may be defined relative to thedimensions of the backing 901. For example, in one embodiment, thegrinding direction 985 may be substantially perpendicular to the lateralaxis 981 of the backing and substantially parallel to the longitudinalaxis 980 of the backing 901. The predetermined orientationcharacteristics of the shaped abrasive particle 902 may define aninitial contact surface of the shaped abrasive particle 902 with aworkpiece. For example, the shaped abrasive particle 902 can includemajor surfaces 963 and 964 and side surfaces 965 and 966, each of whichcan extend between the major surfaces 963 and 964. The predeterminedorientation characteristics of the shaped abrasive particle 902 canposition the particle 902 such that the major surface 963 is configuredto make initial contact with a workpiece before the other surfaces ofthe shaped abrasive particle 902 during a material removal operation.Such an orientation may be considered a major surface orientationrelative to the grinding direction 985. More particularly, the shapedabrasive particle 902 can have a bisecting axis 931 having a particularorientation relative to the grinding direction 985. For example, asillustrated, the vector of the grinding direction 985 and the bisectingaxis 931 are substantially perpendicular to each other. It will beappreciated that, just as any range of predetermined rotationalorientations relative to the backing are contemplated for a shapedabrasive particle, any range of orientations of the shaped abrasiveparticles relative to the grinding direction 985 are contemplated andcan be utilized.

The shaped abrasive particle 903 can have one or more differentpredetermined orientation characteristics as compared to the shapedabrasive particle 902 and the grinding direction 985. As illustrated,the shaped abrasive particle 903 can include major surfaces 991 and 992,each of which can be joined by side surfaces 971 and 972. Moreover, asillustrated, the shaped abrasive particle 903 can have a bisecting axis973 forming a particular angle relative to the vector of the grindingdirection 985. As illustrated, the bisecting axis 973 of the shapedabrasive particle 903 can have a substantially parallel orientation withthe grinding direction 985 such that the angle between the bisectingaxis 973 and the grinding direction 985 is essentially 0 degrees.Accordingly, the predetermined orientation characteristics of the shapedabrasive particle 903 facilitate initial contact of the side surface 972with a workpiece before any of the other surfaces of the shaped abrasiveparticle 903. Such an orientation of the shaped abrasive particle 903may be considered a side surface orientation relative to the grindingdirection 985.

Still, in one non-limiting embodiment, it will be appreciated that anabrasive article can include one or more groups of shaped abrasiveparticles that can be arranged in one or more predetermineddistributions relative to the backing, a grinding direction, and/or eachother. For example, one or more groups of shaped abrasive particles, asdescribed herein, can have a predetermined orientation relative to agrinding direction. Moreover, the abrasive articles herein can have oneor more groups of shaped abrasive particles, each of the groups having adifferent predetermined orientation relative to a grinding direction.Utilization of groups of shaped abrasive particles having differentpredetermined orientations relative to a grinding direction mayfacilitate improved performance of the abrasive article.

FIG. 10 includes a top view illustration of a portion of an abrasivearticle in accordance with an embodiment. In particular, the abrasivearticle 1000 can include a first group 1001 including a plurality ofshaped abrasive particles. As illustrated, the shaped abrasive particlescan be arranged relative to each other one the backing 101 to define apredetermined distribution. More particularly, the predetermineddistribution can be in the form of a pattern 1023 as viewed top-down,and more particularly defining a triangular shaped two-dimensionalarray. As further illustrated, the first group 1001 can be arranged onthe abrasive article 1000 defining a predetermined macro-shape 1031overlying the backing 101. In accordance with an embodiment, themacro-shape 1031 can have a particular two-dimensional shape as viewedtop-down. Some exemplary two-dimensional shapes can include polygons,ellipsoids, numerals, Greek alphabet characters, Latin alphabetcharacters, Russian alphabet characters, Arabic alphabet characters,Kanji characters, complex shapes, irregular shapes, designs, any acombination thereof. In particular instances, the formation of a grouphaving a particular macro-shape may facilitate improved performance ofthe abrasive article.

As further illustrated, the abrasive article 1000 can include a group1004 including a plurality of shaped abrasive particles which can bearranged on the surface of the backing 101 relative to each other todefine a predetermined distribution. Notably, the predetermineddistribution can include an arrangement of the plurality of the shapedabrasive particles that define a pattern 422, and more particularly, agenerally quadrilateral pattern. As illustrated, the group 1004 candefine a macro-shape 1034 on the surface of the abrasive article 1000.In one embodiment, the macro-shape 1034 of the group 1004 can have atwo-dimensional shape as viewed top down, including for example apolygonal shape, and more particularly, a generally quadrilateral(diamond) shape as viewed top down on the surface of the abrasivearticle 1000. In the illustrated embodiment of FIG. 10, the group 1001can have a macro-shape 1031 that is substantially the same as themacro-shape 1034 of the group 1004. However, it will be appreciated thatin other embodiments, various different groups can be used on thesurface of the abrasive article, and more particularly wherein each ofthe different groups has a different macro-shape relative to each other.

As further illustrated, the abrasive article can include groups 1001,1002, 1003, and 1004 which can be separated by channel regions 1021 and1024 extending between the groups 1001-1004. In particular instances,the channel regions 1021 and 1024 can be substantially free of shapedabrasive particles. Moreover, the channel regions 1021 and 1024 may beconfigured to move liquid between the groups 1001-1004 and furtherimprove swarf removal and grinding performance of the abrasive article.Furthermore, in a certain embodiment, the abrasive article 1000 caninclude channel regions 1021 and 1024 extending between groups1001-1004, wherein the channel regions 1021 and 1024 can be patterned onthe surface of the abrasive article 1000. In particular instances, thechannel regions 1021 and 1024 can represent a regular and repeatingarray of features extending along a surface of the abrasive article.

The fixed abrasive articles of the embodiments herein can be utilized invarious material removal operations. For example, fixed abrasivearticles herein can be used in methods of removing material from aworkpiece by moving the fixed abrasive article relative to theworkpiece. The relative movement between the fixed abrasive and theworkpiece can facilitate removal of the material from the surface of theworkpiece. Various workpieces can be modified using the fixed abrasivearticles of the embodiments herein, including but not limited to,workpieces comprising inorganic materials, organic materials, and acombination thereof. In a particular embodiment, the workpiece mayinclude a metal, such as a metal alloy. In one particular instance, theworkpiece can consist essentially of a metal or metal alloy, such asstainless steel.

Many different aspects and embodiments are possible. Some of thoseaspects and embodiments are described herein. After reading thisspecification, skilled artisans will appreciate that those aspects andembodiments are only illustrative and do not limit the scope of thepresent invention. Embodiments may be in accordance with any one or moreof the items as listed below.

Items:

Item 1. A shaped abrasive particle comprising:

a body comprising a first major surface, a second major surface, and aside surface extending between the first major surface and the secondmajor surface, wherein the body comprises a sharpness-shape-strengthfactor (3SF) within a range between about 0.7 and about 1.7 and a ShapeIndex within a range between at least about 0.51 and not greater thanabout 0.99.

Item 2. A shaped abrasive particle comprising:

a body comprising a first major surface, a second major surface, and aside surface extending between the first major surface and the secondmajor surface, wherein the body comprises a Shape Index within a rangebetween at least about 0.51 and not greater than about 0.99 and astrength of not greater than about 600 MPa and at least about 100 MPa.

Item 3. A shaped abrasive particle comprising:

a body comprising a first major surface, a second major surface, and aside surface extending between the first major surface and the secondmajor surface, wherein the body comprises at least four corners, whereineach corner comprises a tip sharpness within a range between not greaterthan about 80 microns and at least about 1 micron, a Shape Index withina range between at least about 0.51 and not greater than about 0.99, andthe body comprises a strength of not greater than about 600 MPa and atleast about 100 MPa.

Item 4. The shaped abrasive particle of any one of items 1, 2, and 3,wherein the body comprises a Shape Index of at least about 0.52, atleast about 0.53, at least about 0.54, at least about 0.55, at leastabout 0.56, at least about 0.57, at least about 0.58, at least about0.59, at least about 0.60, at least about 0.61, at least about 0.62, atleast about 0.63, at least about 0.64, at least about 0.65, at leastabout 0.66, at least about 0.67, at least about 0.68, at least about0.69, at least about 0.70, at least about 0.71, at least about 0.72, atleast about 0.73, at least about 0.74, at least about 0.75, at leastabout 0.76, at least about 0.77, at least about 0.78, at least about0.79, at least about 0.80, at least about 0.81, at least about 0.82, atleast about 0.83, at least about 0.84, at least about 0.85, at leastabout 0.86, at least about 0.87, at least about 0.88, at least about0.89, at least about 0.90, at least about 0.91, at least about 0.92, atleast about 0.93, at least about 0.94, at least about 0.95.

Item 5. The shaped abrasive particle of any one of items 1, 2, and 3,wherein the body comprises a Shape Index of not greater than about 0.98,not greater than about 0.97, not greater than about 0.96, not greaterthan about 0.95, not greater than about 0.94, not greater than about0.93, not greater than about 0.92, not greater than about 0.91, notgreater than about 0.90, not greater than about 0.89, not greater thanabout 0.88, not greater than about 0.87, not greater than about 0.86,not greater than about 0.85, not greater than about 0.84, not greaterthan about 0.83, not greater than about 0.82, not greater than about0.81, not greater than about 0.80, not greater than about 0.79, notgreater than about 0.78, not greater than about 0.77, not greater thanabout 0.76, not greater than about 0.75, not greater than about 0.74,not greater than about 0.73, not greater than about 0.72, not greaterthan about 0.71, not greater than about 0.70, not greater than about0.69, not greater than about 0.68, not greater than about 0.67, notgreater than about 0.66, not greater than about 0.65, not greater thanabout 0.64, not greater than about 0.63, not greater than about 0.62,not greater than about 0.61, not greater than about 0.60, not greaterthan about 0.59, not greater than about 0.58, not greater than about0.57, not greater than about 0.56, not greater than about 0.55, notgreater than about 0.54.

Item 6. The shaped abrasive particle of any one of items 2, and 3,wherein the body comprises a sharpness-shape-strength factor (3SF)within a range between about 0.7 and about 1.7.

Item 7. The shaped abrasive particle of any one of items 1 and 6,wherein the body has a 3SF of at least about 0.72, at least about 0.75,at least about 0.78, at least about 0.8, at least about 0.82, at leastabout 0.85, at least about 0.88, at least about 0.90, at least about0.92, at least about 0.95, at least about 0.98.

Item 8. The shaped abrasive particle of any one of items 1 and 6,wherein the body has a 3SF of not greater than about 1.68, not greaterthan about 1.65, not greater than about 1.62, not greater than about1.6, not greater than about 1.58, not greater than about 1.55, notgreater than about 1.52, not greater than about 1.5, not greater thanabout 1.48, not greater than about 1.45, not greater than about 1.42,not greater than about 1.4, not greater than about 1.38, not greaterthan about 1.35, not greater than about 1.32, not greater than about1.3, not greater than about 1.28, not greater than about 1.25, notgreater than about 1.22, not greater than about 1.2, not greater thanabout 1.18, not greater than about 1.15, not greater than about 1.12,not greater than about 1.1.

Item 9. The shaped abrasive particle of item 1, wherein the bodycomprises a strength of not greater than about 600 MPa and at leastabout 100 MPa.

Item 10. The shaped abrasive particle of any one of items 2, 3, and 9,wherein the body comprises a strength of not greater than about 590 MPa,not greater than about 580 MPa, not greater than about 570 MPa, notgreater than about 560 MPa, not greater than about 550 MPa, not greaterthan about 540 MPa, not greater than about 530 MPa, not greater thanabout 520 MPa, not greater than about 510 MPa, not greater than about500 MPa, not greater than about 490 MPa, not greater than about 480 MPa,not greater than about 470 MPa, not greater than about 460 MPa, notgreater than about 450 MPa, not greater than about 440 MPa, not greaterthan about 430 MPa, not greater than about 420 MPa, not greater thanabout 410 MPa, not greater than about 400 MPa, not greater than about390 MPa, not greater than about 380 MPa, not greater than about 370 MPa,not greater than about 360 MPa, not greater than about 350 MPa, notgreater than about 340 MPa, not greater than about 330 MPa, not greaterthan about 320 MPa, not greater than about 310 MPa, not greater thanabout 300 MPa, not greater than about 290 MPa, not greater than about280 MPa, not greater than about 270 MPa, not greater than about 260 MPa,not greater than about 250 MPa, not greater than about 240 MPa, notgreater than about 230 MPa, not greater than about 220 MPa, not greaterthan about 210 MPa, not greater than about 200 MPa.

Item 11. The shaped abrasive particle of any one of items 2, 3, and 9,wherein the body comprises a strength of at least about 110 MPa, atleast about 120 MPa, at least about 130 MPa, at least about 140 MPa, atleast about 150 MPa, at least about 160 MPa, at least about 170 MPa, atleast about 180 MPa, at least about 190 MPa, at least about 200 MPa, atleast about 210 MPa, at least about 220 MPa, at least about 230 MPa, atleast about 240 MPa, at least about 250 MPa, at least about 260 MPa, atleast about 270 MPa, at least about 280 MPa, at least about 290 MPa, atleast about 300 MPa, at least about 310 MPa, at least about 320 MPa, atleast about 330 MPa, at least about 340 MPa, at least about 350 MPa, atleast about 360 MPa, at least about 370 MPa, at least about 380 MPa, atleast about 390 MPa, at least about 400 MPa, at least about 410 MPa, atleast about 420 MPa, at least about 430 MPa, at least about 440 MPa, atleast about 450 MPa, at least about 460 MPa, at least about 470 MPa, atleast about 480 MPa, at least about 490 MPa, at least about 500 MPa.

Item 12. The shaped abrasive particle of any one of items 1 and 2,wherein the body comprises a tip sharpness within a range between notgreater than about 80 microns and at least about 1 micron.

Item 13. The shaped abrasive particle of any one of items 3 and 12,wherein the body comprises a tip sharpness of not greater than about 78microns, not greater than about 76 microns, not greater than about 74microns, not greater than about 72 microns, not greater than about 70microns, not greater than about 68 microns, not greater than about 66microns, not greater than about 64 microns, not greater than about 62microns, not greater than about 60 microns, not greater than about 58microns, not greater than about 56 microns, not greater than about 54microns, not greater than about 52 microns, not greater than about 50microns, not greater than about 48 microns, not greater than about 46microns, not greater than about 44 microns, not greater than about 42microns, not greater than about 40 microns, not greater than about 38microns, not greater than about 36 microns, not greater than about 34microns, not greater than about 32 microns, not greater than about 30microns, not greater than about 38 microns, not greater than about 36microns, not greater than about 34 microns, not greater than about 32microns, not greater than about 30 microns, not greater than about 28microns, not greater than about 26 microns, not greater than about 24microns, not greater than about 22 microns, not greater than about 20microns, not greater than about 18 microns, not greater than about 16microns, not greater than about 14 microns, not greater than about 12microns, not greater than about 10 microns.

Item 14. The shaped abrasive particle of any one of items 3 and 12,wherein the body comprises a tip sharpness of at least about 2 microns,at least about 4 microns, at least about 6 microns, at least about 8microns, at least about 10 microns, at least about 12 microns, at leastabout 14 microns, at least about 16 microns, at least about 18 microns,at least about 20 microns, at least about 22 microns, at least about 24microns, at least about 26 microns, at least about 28 microns, at leastabout 30 microns, at least about 32 microns, at least about 34 microns,at least about 36 microns, at least about 38 microns, at least about 40microns, at least about 42 microns, at least about 44 microns, at leastabout 46 microns, at least about 48 microns, at least about 50 microns,at least about 52 microns, at least about 54 microns, at least about 56microns, at least about 58 microns, at least about 60 microns, at leastabout 62 microns, at least about 64 microns, at least about 66 microns,at least about 68 microns, at least about 70 microns.

Item 15. The shaped abrasive particle of any one of items 1, 2, and 3,wherein the body comprises an additive, wherein the additive comprisesan oxide, wherein the additive comprises a metal element, wherein theadditive comprises a rare-earth element.

Item 16. The shaped abrasive article of item 15, wherein the additivecomprises a dopant material, wherein the dopant material includes anelement selected from the group consisting of an alkali element, analkaline earth element, a rare earth element, a transition metalelement, and a combination thereof, wherein the dopant materialcomprises an element selected from the group consisting of hafnium,zirconium, niobium, tantalum, molybdenum, vanadium, lithium, sodium,potassium, magnesium, calcium, strontium, barium, scandium, yttrium,lanthanum, cesium, praseodymium, chromium, cobalt, iron, germanium,manganese, nickel, titanium, zinc, and a combination thereof.

Item 17. The shaped abrasive article of item 16, wherein the dopantcomprises MgO, and wherein the body comprises a content of MgO of atleast about 0.5 wt %, at least about 0.6 wt %, at least about 0.7 wt %,at least about 0.8 wt %, at least about 0.9 wt %, at least about 1 wt %,at least about 1.1 wt %, at least about 1.2 wt %, at least about 1.3 wt%, at least about 1.4 wt %, at least about 1.5 wt %, at least about 1.6wt %, at least about 1.7 wt %, at least about 1.8 wt %, at least about1.9 wt %, at least about 2 wt %, at least about 2.1 wt %, at least about2.2 wt %, at least about 2.3 wt %, at least about 2.4 wt %, at leastabout 2.5 wt %.

Item 18. The shaped abrasive article of item 17, wherein the bodycomprises a content of MgO of not greater than about 8 wt %, not greaterthan about 7 wt %, not greater than about 6 wt %, not greater than about5 wt %, not greater than about 4.9 wt %, not greater than about 4.8 wt%, not greater than about 4.7 wt %, not greater than about 4.6 wt %, notgreater than about 4.5 wt %, not greater than about 4.4 wt %, notgreater than about 4.3 wt %, not greater than about 4.2 wt %, notgreater than about 4.1 wt %, not greater than about 4 wt %, not greaterthan about 3.9 wt %, not greater than about 3.8 wt %, not greater thanabout 3.7 wt %, not greater than about 3.6 wt %, not greater than about3.5 wt %, not greater than about 3.4 wt %, not greater than about 3.3 wt%, not greater than about 3.2 wt %, not greater than about 3.1 wt %, notgreater than about 3 wt %, not greater than about 2.9 wt %, not greaterthan about 2.8 wt %, not greater than about 2.7 wt %, not greater thanabout 2.6 wt %, not greater than about 2.5 wt %.

Item 19. The shaped abrasive particle of any one of items 1, 2, and 3,wherein the body comprises at least about 95 wt % alumina for the totalweight of the body, at least about 95.1 wt %, at least about 95.2 wt %,at least about 95.3 wt %, at least about 95.4 wt %, at least about 95.5wt %, at least about 95.6 wt %, at least about 95.7 wt %, at least about95.8 wt %, at least about 95.9 wt %, at least about 96 wt %, at leastabout 96.1 wt %, at least about 96.2 wt %, at least about 96.3 wt %, atleast about 96.4 wt %, at least about 96.5 wt %, at least about 96.6 wt%, at least about 96.7 wt %, at least about 96.8 wt %, at least about96.9 wt %, at least about 97 wt %, at least about 97.1 wt %, at leastabout 97.2 wt %, at least about 975.3 wt %, at least about 97.4 wt %, atleast about 97.5 wt %.

Item 20. The shaped abrasive particle of any one of items 1, 2, and 3,wherein the body comprises not greater than about 99.5 wt % alumina forthe total weight of the body, not greater than about 99.4 wt %, notgreater than about 99.3 wt %, not greater than about 99.2 wt %, notgreater than about 99.1 wt %, not greater than about 99 wt %, notgreater than about 98.9 wt %, not greater than about 98.8 wt %, notgreater than about 98.7 wt %, not greater than about 98.6 wt %, notgreater than about 98.5 wt %, not greater than about 98.4 wt %, notgreater than about 98.3 wt %, not greater than about 98.2 wt %, notgreater than about 98.1 wt %, not greater than about 98 wt %, notgreater than about 97.9 wt %, not greater than about 97.8 wt %, notgreater than about 97.7 wt %, not greater than about 97.6 wt %, notgreater than about 97.5 wt %.

Item 21. The shaped abrasive particle of any one of items 1, 2, and 3,wherein the body consists essentially of alumina and MgO.

Item 22. The shaped abrasive particle of any one of items 1, 2, and 3,wherein the body consists essentially of alumina.

Item 23. The shaped abrasive particle of any one of items 1, 2, and 3,wherein the body comprises a polycrystalline material includingcrystalline grains, wherein the average grain size is not greater thanabout 1 micron, not greater than about 0.9 microns, not greater thanabout 0.8 microns, not greater than about 0.7 microns, not greater thanabout 0.6 microns.

Item 24. The shaped abrasive particle of item 23, wherein the averagegrain size is at least about 0.01 microns, at least about 0.05 microns,at least about 0.06 microns, at least about 0.07 microns, at least about0.08 microns, at least about 0.09 microns, at least about 0.1 microns,at least about 0.12 microns, at least about 0.15 microns, at least about0.17 microns, at least about 0.2 microns.

Item 25. The shaped abrasive particle of any one of items 1, 2, and 3,wherein the body is essentially free of a binder, wherein the body isessentially free of an organic material, wherein the body is essentiallyfree of rare earth elements, wherein the body is essentially free ofiron.

Item 26. The shaped abrasive particle of any one of items 1, 2, and 3,wherein the body is formed from a seeded sol gel.

Item 27. The shaped abrasive particle of any one of items 1, 2, and 3,wherein the body comprises a primary aspect ratio of width:length of atleast about 1:1 and not greater than about 1:10.

Item 28. The shaped abrasive particle of any one of items 1, 2, and 3,wherein the body comprises a secondary aspect ratio defined by a ratioof width:height within a range between about 5:1 and about 1:1.

Item 29. The shaped abrasive particle of any one of items 1, 2, and 3,wherein the body comprises a tertiary aspect ratio defined by a ratio oflength:height within a range between about 6:1 and about 1:1.

Item 30. The shaped abrasive particle of any one of items 1, 2, and 3,wherein the body comprises a polycrystalline material, wherein thepolycrystalline material comprises grains, wherein the grains areselected from the group of materials consisting of nitrides, oxides,carbides, borides, oxynitrides, diamond, and a combination thereof,wherein the grains comprise an oxide selected from the group of oxidesconsisting of aluminum oxide, zirconium oxide, titanium oxide, yttriumoxide, chromium oxide, strontium oxide, silicon oxide, and a combinationthereof, wherein the grains comprise alumina, wherein the grains consistessentially of alumina.

Item 31. The shaped abrasive particle of any one of items 1, 2, and 3,wherein the body comprises a two-dimensional polygonal shape as viewedin a plane defined by a length and width, wherein the body comprises atwo-dimensional shape selected from the group consisting ofquadrilateral, rectangular, trapezoidal, pentagonal, hexagonal,heptagonal, octagonal, and a combination thereof, wherein the bodycomprises a two-dimensional shape as viewed in a plane defined by alength and a width of the body selected from the group consisting ofellipsoids, Greek alphabet characters, Latin alphabet characters,Russian alphabet characters, and a combination thereof.

Item 32. The shaped abrasive particle of any one of items 1, 2, and 3,wherein the body comprises a two-dimensional polygonal shape as viewedin a plane defined by a length and width, wherein the body comprises ashape selected from the group consisting of an irregular quadrilateral,an irregular rectangle, an irregular trapezoid, an irregular pentagon,an irregular hexagon, an irregular heptagon, an irregular octagon, anirregular nonagon, an irregular decagon, and a combination thereof.

Item 33. The shaped abrasive particle of any one of items 1, 2, and 3,wherein the body comprises a two-dimensional polygonal shape as viewedin a plane defined by a length and width, wherein the body comprises atwo dimensional shape having at least 4 exterior points, at least 5exterior points, at least 6 exterior points, at least 7 exterior points,at least 8 exterior points, at least 9 exterior points.

Item 34. The shaped abrasive particle of any one of items 1, 2, and 3,wherein the body is coupled to a substrate as part of a fixed abrasive,wherein the fixed abrasive article is selected from the group consistingof a bonded abrasive article, a coated abrasive article, and acombination thereof.

Item 35. The shaped abrasive particle of item 34, wherein the substrateis a backing, wherein the backing comprises a woven material, whereinthe backing comprises a non-woven material, wherein the backingcomprises an organic material, wherein the backing comprises a polymer,wherein the backing comprises a material selected from the groupconsisting of cloth, paper, film, fabric, fleeced fabric, vulcanizedfiber, woven material, non-woven material, webbing, polymer, resin,phenolic resin, phenolic-latex resin, epoxy resin, polyester resin, ureaformaldehyde resin, polyester, polyurethane, polypropylene, polyimides,and a combination thereof.

Item 36. The shaped abrasive particle of item 34, wherein the shapedabrasive particle is part of a plurality of a first type of shapedabrasive particles, wherein a majority of the first type of shapedabrasive particles are coupled to the backing in an open coat, whereinthe open coat comprises a coating density of not greater than about 70particles/cm2, not greater than about 65 particles/cm2, not greater thanabout 60 particles/cm2, not greater than about 55 particles/cm2, notgreater than about 50 particles/cm2, at least about 5 particles/cm2, atleast about 10 particles/cm2.

Item 37. The shaped abrasive particle of item 34, wherein the shapedabrasive particle is part of a plurality of a first type of shapedabrasive particles, wherein a majority of the first type of shapedabrasive particles are coupled to the backing in a closed coat, whereinhaving a closed coat of the blend of shaped abrasive particles on abacking, wherein the closed coat comprises a coating density of at leastabout 75 particles/cm2, at least about 80 particles/cm2, at least about85 particles/cm2, at least about 90 particles/cm2, at least about 100particles/cm2.

Item 38. The shaped abrasive particle of item 34, wherein the shapedabrasive particle is part of a blend including a plurality of a firsttype of shaped abrasive particles and a third type of abrasive particle,wherein the third type of abrasive particle comprises a shaped abrasiveparticle, wherein the third type of abrasive particle comprises adiluent type of abrasive particle, wherein the diluent type of abrasiveparticle comprises an irregular shape.

Item 39. The shaped abrasive particle of item 38, wherein the blend ofabrasive particles comprises a plurality of shaped abrasive particles,and wherein each shaped abrasive particle of the plurality of shapedabrasive particles is arranged in a controlled orientation relative to abacking, the controlled orientation including at least one of apredetermined rotational orientation, a predetermined lateralorientation, and a predetermined longitudinal orientation.

Item 40. The shaped abrasive particle of item 35, wherein the backingcomprises an additive selected from the group consisting of catalysts,coupling agents, currants, anti-static agents, suspending agents,anti-loading agents, lubricants, wetting agents, dyes, fillers,viscosity modifiers, dispersants, defoamers, and grinding agents.

Item 41. The shaped abrasive particle of item 35, further comprising anadhesive layer overlying the backing, wherein the adhesive layercomprises a make coat, wherein the make coat overlies the backing,wherein the make coat is bonded directly to a portion of the backing,wherein the make coat comprises an organic material, wherein the makecoat comprises a polymeric material, wherein the make coat comprises amaterial selected from the group consisting of polyesters, epoxy resins,polyurethanes, polyamides, polyacrylates, polymethacrylates, poly vinylchlorides, polyethylene, polysiloxane, silicones, cellulose acetates,nitrocellulose, natural rubber, starch, shellac, and a combinationthereof.

Item 42. The shaped abrasive particle of item 40, wherein the adhesivelayer comprises a size coat, wherein the size coat overlies a portion ofthe plurality of shaped abrasive particles, wherein the size coatoverlies a make coat, wherein the size coat is bonded directly to aportion of the plurality of shaped abrasive particles, wherein the sizecoat comprises an organic material, wherein the size coat comprises apolymeric material, wherein the size coat comprises a material selectedfrom the group consisting of polyesters, epoxy resins, polyurethanes,polyamides, polyacrylates, polymethacrylates, polyvinyl chlorides,polyethylene, polysiloxane, silicones, cellulose acetates,nitrocellulose, natural rubber, starch, shellac, and a combinationthereof.

Item 43. The shaped abrasive particle of any one of items 1, 2, and 3,wherein the body comprises a length (l), a width (w), and a height (hi),wherein the length>width, the length>height, and the width>height.

Item 44. The shaped abrasive particle of item 43, wherein the height (h)is at least about 100 microns, at least about 110 microns, at leastabout 120 microns, at least about 150 microns, at least about 175microns, at least about 200 microns, at least about 225 microns, atleast about 250 microns, at least about 275 microns, or even at leastabout 300 microns, and not greater than about 3 mm, such as not greaterthan about 2 mm, not greater than about 1.5 mm, not greater than about 1mm, or even not greater than about 800 microns, not greater than about600 microns, not greater than about 500 microns, not greater than about475 microns, not greater than about 450 microns, not greater than about425 microns, not greater than about 400 microns, not greater than about375 microns, not greater than about 350 microns, not greater than about325 microns, not greater than about 300 microns, not greater than about275 microns, or even not greater than about 250 microns.

Item 45. The shaped abrasive particle of item 44, wherein the width isat least about 600 microns, at least about 700 microns, at least about800 microns, at least about 900 microns, and not greater than about 4mm, not greater than about 3 mm, not greater than about 2.5 mm, notgreater than about 2 mm.

Item 46. The shaped abrasive particle of any one of items 1, 2, and 3,wherein the body comprises a percent flashing of not greater than about40%, not greater than about 35%, not greater than about 30%, not greaterthan about 25%, not greater than about 20%, not greater than about 18%,not greater than about 15%, not greater than about 12%, not greater thanabout 10%, not greater than about 8%, not greater than about 6%, notgreater than about 4%.

Item 47. The shaped abrasive particle of any one of items 1, 2, and 3,wherein the body comprises a dishing value (d) of not greater than about2, not greater than about 1.9, not greater than about 1.8, not greaterthan about 1.7, not greater than about 1.6, not greater than about 1.5,not greater than about 1.2, and at least about 0.9, at least about 1.0.

Item 48. The shaped abrasive particle of any one of items 1, 2, and 3,wherein the shaped abrasive particle is part of a plurality of a firsttype of shaped abrasive particles, wherein a majority of the first typeof shaped abrasive particles are coupled to the backing in a sideorientation, wherein at least about 55% of the shaped abrasive particlesof the plurality of shaped abrasive particles are coupled to the backingin a side orientation, at least about 60%, at least about 65%, at leastabout 70%, at least about 75%, at least about 77%, at least about 80%,and not greater than about 99%, not greater than about 95%, not greaterthan about 90%, not greater than about 85%.

Item 49. A method of making a shaped abrasive particle comprisingforming a body of a shaped abrasive comprising at least one of:

i) selecting a material having a predetermined strength and forming thebody of the shaped abrasive particle with a predetermined tip sharpnessand predetermined Shape Index based upon the predetermined strength;

ii) selecting a predetermined Shape Index of the body of the shapedabrasive particle and forming the body with the predetermined tipsharpness and the predetermined strength based upon the predeterminedShape Index; and

iii) selecting a predetermined tip sharpness of a body of the shapedabrasive particle and forming the body of a shaped abrasive particlewith a predetermined Shape Index and a predetermined strength based uponthe predetermined tip sharpness.

Item 50. The method of item 49, wherein forming comprises controllingthe interrelationship of grain features selected from the group ofpredetermined tip sharpness, predetermined Shape Index, andpredetermined strength of the body of the shaped abrasive particle toinfluence a self-sharpening behavior of the shaped abrasive particle.

Item 51. The method of item 50, wherein controlling theinterrelationship of grain features comprises forming the body with asharpness-shape-strength factor (3SF) within a range between about 0.7and about 1.7.

Item 52. The method of item 49, wherein forming comprises a methodselected from the group consisting of depositing, printing, extruding,molding, casting, pressing, punching, sectioning, and a combinationthereof.

Item 53. The method of item 49, wherein the predetermined Shape Index iswithin a range between at least about 0.51 and not greater than about0.99.

Item 54. The method of item 49, wherein the predetermined tip sharpnessis within a range between not greater than about 80 microns and at leastabout 1 micron.

Item 55. The method of item 49, wherein the predetermined strength iswithin a range between not greater than about 600 MPa and at least about100 MPa.

Item 56. A shaped abrasive particle comprising:

a body comprising a first major surface, a second major surface, and aside surface extending between the first major surface and the secondmajor surface, wherein the body comprises an oblique, truncated shape.

Item 57. The shaped abrasive particle of item 56, wherein the sidesurface comprises a first side section and a first oblique side section,wherein the first side section and the first oblique side section arejoined at a first oblique angle (Ao1) having an obtuse value.

Item 58. The shaped abrasive particle of item 57, wherein the obtuseangle has a value of at least about 92 degrees, at least about 94degrees, at least about 96 degrees, at least about 98 degrees, at leastabout 100 degrees, at least about 102 degrees, at least about 104degrees, at least about 106 degrees, at least about 108 degrees, atleast about 110 degrees, at least about 112 degrees, at least about 124degrees, at least about 126 degrees, at least about 128 degrees, atleast about 120 degrees, at least about 122 degrees, at least about 124degrees, at least about 126 degrees, at least about 128 degrees, atleast about 130 degrees, at least about 132 degrees, at least about 134degrees, at least about 136 degrees, at least about 138 degrees, atleast about 140 degrees.

Item 59. The shaped abrasive particle of item 57, wherein the obtuseangle is not greater than about 176 degrees, not greater than about 174degrees, not greater than about 172 degrees, not greater than about 170degrees, not greater than about 168 degrees, not greater than about 166degrees, not greater than about 164 degrees, not greater than about 162degrees, not greater than about 160 degrees, not greater than about 158degrees, not greater than about 156 degrees, not greater than about 154degrees, not greater than about 152 degrees, not greater than about 150degrees, not greater than about 148 degrees, not greater than about 146degrees, not greater than about 144 degrees, not greater than about 142degrees, not greater than about 140 degrees.

Item 60. The shaped abrasive particle of item 56, wherein the first sidesection comprises a first side section length (Lss1) and the firstoblique side section has a length (Los1), and where the length of thefirst oblique side section is different than the length of the firstside section.

Item 61. The shaped abrasive particle of item 60, wherein Los1>Lss1.

Item 62. The shaped abrasive particle of item 60, wherein Lss1>Los1.

Item 63. The shaped abrasive particle of item 60, further comprising alength factor (Los1/Lss1) of not greater than about 1, not greater thanabout 0.95, not greater than about 0.9, not greater than about 0.85, notgreater than about 0.8, not greater than about 0.75, not greater thanabout 0.7, not greater than about 0.65, not greater than about 0.6, notgreater than about 0.55, not greater than about 0.5, not greater thanabout 0.45, not greater than about 0.4, not great not greater than about0.35, not greater than about 0.3, not greater than about 0.35, notgreater than about 0.3, not greater than about 0.25, not greater thanabout 0.2, not greater than about 0.15, not greater than about 0.1, notgreater than about 0.05.

Item 64. The shaped abrasive particle of item 63, wherein the lengthfactor (Los1/Lss1) is at least about 0.05, at least about 0.1, at leastabout 0.15, at least about 0.2, at least about 0.25, at least about 0.3,at least about 0.35, at least about 0.4, at least about 0.45, at leastabout 0.5, at least about 0.55, at least about 0.6, at least about 0.65,at least about 0.7, at least about 0.75, at least about 0.8, at leastabout 0.85, at least about 0.9, at least about 0.95.

Item 65. The shaped abrasive particle of item 60, further comprising alength factor (Lss1/Los1) of not greater than about 1, not greater thanabout 0.95, not greater than about 0.9, not greater than about 0.85, notgreater than about 0.8, not greater than about 0.75, not greater thanabout 0.7, not greater than about 0.65, not greater than about 0.6, notgreater than about 0.55, not greater than about 0.5, not greater thanabout 0.45, not greater than about 0.4, not great not greater than about0.35, not greater than about 0.3, not greater than about 0.35, notgreater than about 0.3, not greater than about 0.25, not greater thanabout 0.2, not greater than about 0.15, not greater than about 0.1, notgreater than about 0.05.

Item 66. The shaped abrasive particle of item 65, wherein the lengthfactor (Lss1/Los1) is at least about 0.05, at least about 0.1, at leastabout 0.15, at least about 0.2, at least about 0.25, at least about 0.3,at least about 0.35, at least about 0.4, at least about 0.45, at leastabout 0.5, at least about 0.55, at least about 0.6, at least about 0.65,at least about 0.7, at least about 0.75, at least about 0.8, at leastabout 0.85, at least about 0.9, at least about 0.95.

Item 67. The shaped abrasive particle of item 57, wherein the sidesurface further comprises a second side section and a second obliqueside section, wherein the second side section is coupled to the firstoblique section at a first external corner, wherein the first externalcorner defines a first external corner angle (Aec1) and wherein thefirst external corner angle is different than a value of the firstoblique angle (Ao1).

Item 68. The shaped abrasive particle of item 67, wherein the firstexternal corner angle is less than the value of the first oblique angle.

Item 69. The shaped abrasive particle of item 67, wherein the firstexternal corner angle is at least about 50 degrees, at least about 55degrees, at least about 60 degrees, at least about 65 degrees, at leastabout 70 degrees, at least about 80 degrees, at least about 85 degrees.

Item 70. The shaped abrasive particle of item 69, wherein the firstexternal corner angle is not greater than about 130 degrees, not greaterthan about 125 degrees, not greater than about 120 degrees, not greaterthan about 115 degrees, not greater than about 110 degrees, not greaterthan about 105 degrees, not greater than about 100 degrees, not greaterthan about 95 degrees, not greater than about 94 degrees, not greaterthan about 93 degrees.

Item 71. The shaped abrasive particle of item 67, further comprising afirst angle factor (Aec1/Ao1) of not greater than about 1, not greaterthan about 0.95, not greater than about 0.9, not greater than about0.85, not greater than about 0.8, not greater than about 0.75, notgreater than about 0.7, not greater than about 0.65, not greater thanabout 0.6, not greater than about 0.55, not greater than about 0.5, notgreater than about 0.45, not greater than about 0.4, not great notgreater than about 0.35, not greater than about 0.3, not greater thanabout 0.35, not greater than about 0.3, not greater than about 0.25, notgreater than about 0.2, not greater than about 0.15, not greater thanabout 0.1, not greater than about 0.05.

Item 72. The shaped abrasive particle of item 71, wherein the firstangle factor (Aec1/Ao1) is at least about 0.05, at least about 0.1, atleast about 0.15, at least about 0.2, at least about 0.25, at leastabout 0.3, at least about 0.35, at least about 0.4, at least about 0.45,at least about 0.5, at least about 0.55, at least about 0.6, at leastabout 0.65, at least about 0.7, at least about 0.75, at least about 0.8,at least about 0.85, at least about 0.9, at least about 0.95.

Item 73. The shaped abrasive particle of item 68, wherein the firstexternal corner angle is substantially perpendicular.

Item 74. The shaped abrasive particle of item 67, wherein the secondside section comprises a second side section length (Lss2) and thesecond oblique side section has a length (Los2), and where the length ofthe second oblique side section is different than the length of thesecond side section.

Item 75. The shaped abrasive particle of item 74, wherein Lss2 isdifferent than Lss1.

Item 76. The shaped abrasive particle of item 74, wherein Lss2 isessentially the same as Lss1.

Item 77. The shaped abrasive particle of item 74, wherein Los2 isdifferent than Los1.

Item 78. The shaped abrasive particle of item 74, wherein Los2 isessentially the same as Los1.

Item 79. The shaped abrasive particle of item 74, wherein Los2>Lss2.

Item 80. The shaped abrasive particle of item 74, wherein Lss2>Los2.

Item 81. The shaped abrasive particle of item 74, further comprising alength factor (Los2/Lss2) of not greater than about 1, not greater thanabout 0.95, not greater than about 0.9, not greater than about 0.85, notgreater than about 0.8, not greater than about 0.75, not greater thanabout 0.7, not greater than about 0.65, not greater than about 0.6, notgreater than about 0.55, not greater than about 0.5, not greater thanabout 0.45, not greater than about 0.4, not great not greater than about0.35, not greater than about 0.3, not greater than about 0.35, notgreater than about 0.3, not greater than about 0.25, not greater thanabout 0.2, not greater than about 0.15, not greater than about 0.1, notgreater than about 0.05.

Item 82. The shaped abrasive particle of item 81, wherein the lengthfactor (Los2/Lss2) is at least about 0.05, at least about 0.1, at leastabout 0.15, at least about 0.2, at least about 0.25, at least about 0.3,at least about 0.35, at least about 0.4, at least about 0.45, at leastabout 0.5, at least about 0.55, at least about 0.6, at least about 0.65,at least about 0.7, at least about 0.75, at least about 0.8, at leastabout 0.85, at least about 0.9, at least about 0.95.

Item 83. The shaped abrasive particle of item 74, further comprising alength factor (Lss2/Los2) of not greater than about 1, not greater thanabout 0.95, not greater than about 0.9, not greater than about 0.85, notgreater than about 0.8, not greater than about 0.75, not greater thanabout 0.7, not greater than about 0.65, not greater than about 0.6, notgreater than about 0.55, not greater than about 0.5, not greater thanabout 0.45, not greater than about 0.4, not great not greater than about0.35, not greater than about 0.3, not greater than about 0.35, notgreater than about 0.3, not greater than about 0.25, not greater thanabout 0.2, not greater than about 0.15, not greater than about 0.1, notgreater than about 0.05.

Item 84. The shaped abrasive particle of item 83, wherein the lengthfactor (Lss2/Los2) is at least about 0.05, at least about 0.1, at leastabout 0.15, at least about 0.2, at least about 0.25, at least about 0.3,at least about 0.35, at least about 0.4, at least about 0.45, at leastabout 0.5, at least about 0.55, at least about 0.6, at least about 0.65,at least about 0.7, at least about 0.75, at least about 0.8, at leastabout 0.85, at least about 0.9, at least about 0.95.

Item 85. The shaped abrasive particle of item 67, wherein the sidesurface further comprises a third side section and a third oblique sidesection, wherein the third side section is coupled to the second obliquesection at a second external corner.

Item 86. The shaped abrasive particle of item 85, wherein the secondexternal corner defines a second external corner angle (Aec2) andwherein the second external corner angle is different than a value ofthe second oblique angle (Ao1).

Item 87. The shaped abrasive particle of item 86, wherein the secondexternal corner angle (Aec2) is less than the value of the secondoblique angle (Ao2).

Item 88. The shaped abrasive particle of item 86, wherein the secondexternal corner angle is at least about 50 degrees, at least about 55degrees, at least about 60 degrees, at least about 65 degrees, at leastabout 70 degrees, at least about 80 degrees, at least about 85 degrees.

Item 89. The shaped abrasive particle of item 86, wherein the secondexternal corner angle is not greater than about 130 degrees, not greaterthan about 125 degrees, not greater than about 120 degrees, not greaterthan about 115 degrees, not greater than about 110 degrees, not greaterthan about 105 degrees, not greater than about 100 degrees, not greaterthan about 95 degrees, not greater than about 94 degrees, not greaterthan about 93 degrees.

Item 90. The shaped abrasive particle of item 86, further comprising asecond angle factor (Aec2/Ao2) of not greater than about 1, not greaterthan about 0.95, not greater than about 0.9, not greater than about0.85, not greater than about 0.8, not greater than about 0.75, notgreater than about 0.7, not greater than about 0.65, not greater thanabout 0.6, not greater than about 0.55, not greater than about 0.5, notgreater than about 0.45, not greater than about 0.4, not great notgreater than about 0.35, not greater than about 0.3, not greater thanabout 0.35, not greater than about 0.3, not greater than about 0.25, notgreater than about 0.2, not greater than about 0.15, not greater thanabout 0.1, not greater than about 0.05.

Item 91. The shaped abrasive particle of item 90, wherein the secondangle factor (Aec2/Ao2) is at least about 0.05, at least about 0.1, atleast about 0.15, at least about 0.2, at least about 0.25, at leastabout 0.3, at least about 0.35, at least about 0.4, at least about 0.45,at least about 0.5, at least about 0.55, at least about 0.6, at leastabout 0.65, at least about 0.7, at least about 0.75, at least about 0.8,at least about 0.85, at least about 0.9, at least about 0.95.

Item 92. The shaped abrasive particle of item 86, wherein the secondexternal corner angle (Aec2) is substantially perpendicular.

Item 93. The shaped abrasive particle of item 85, wherein the thirdoblique side section is coupled to the first side section at a thirdexternal corner, and wherein the third external corner defines a thirdexternal corner angle (Aec3), and wherein the third external cornerangle is different than a value of a third oblique angle (Ao3).

Item 94. The shaped abrasive particle of item 93, wherein the thirdexternal corner angle (Aec3) is less than the value of the secondoblique angle (Ao3).

Item 95. The shaped abrasive particle of item 93, wherein the thirdexternal corner angle (Aec3) is at least about 50 degrees, at leastabout 55 degrees, at least about 60 degrees, at least about 65 degrees,at least about 70 degrees, at least about 80 degrees, at least about 85degrees.

Item 96. The shaped abrasive particle of item 93, wherein the thirdexternal corner angle (Aec3) is not greater than about 130 degrees, notgreater than about 125 degrees, not greater than about 120 degrees, notgreater than about 115 degrees, not greater than about 110 degrees, notgreater than about 105 degrees, not greater than about 100 degrees, notgreater than about 95 degrees, not greater than about 94 degrees, notgreater than about 93 degrees.

Item 97. The shaped abrasive particle of item 93, wherein the thirdexternal corner angle (Aec3) is substantially perpendicular.

Item 98. The shaped abrasive particle of item 93, further comprising asecond angle factor (Aec2/Ao2) of not greater than about 1, not greaterthan about 0.95, not greater than about 0.9, not greater than about0.85, not greater than about 0.8, not greater than about 0.75, notgreater than about 0.7, not greater than about 0.65, not greater thanabout 0.6, not greater than about 0.55, not greater than about 0.5, notgreater than about 0.45, not greater than about 0.4, not great notgreater than about 0.35, not greater than about 0.3, not greater thanabout 0.35, not greater than about 0.3, not greater than about 0.25, notgreater than about 0.2, not greater than about 0.15, not greater thanabout 0.1, not greater than about 0.05.

Item 99. The shaped abrasive particle of item 98, wherein the secondangle factor (Aec2/Ao2) is at least about 0.05, at least about 0.1, atleast about 0.15, at least about 0.2, at least about 0.25, at leastabout 0.3, at least about 0.35, at least about 0.4, at least about 0.45,at least about 0.5, at least about 0.55, at least about 0.6, at leastabout 0.65, at least about 0.7, at least about 0.75, at least about 0.8,at least about 0.85, at least about 0.9, at least about 0.95.

Item 100. The shaped abrasive particle of item 85, wherein the thirdside section comprises a third side section length (Lss3) and the thirdoblique side section has a length (Los3), and where the length of thethird oblique side section is different than the length of the thirdside section.

Item 101. The shaped abrasive particle of item 100, wherein Lss3 isdifferent than Lss2.

Item 102. The shaped abrasive particle of item 100, wherein Lss3 isdifferent than Lss1.

Item 103. The shaped abrasive particle of item 100, wherein Lss3 isessentially the same as Lss1.

Item 104. The shaped abrasive particle of item 100, wherein Lss3 isessentially the same as Lss2.

Item 105. The shaped abrasive particle of item 100, wherein Los3 isdifferent than Los1.

Item 106. The shaped abrasive particle of item 100, wherein Los3 isdifferent than Los2.

Item 107. The shaped abrasive particle of item 100, wherein Los3 isessentially the same as Los1.

Item 108. The shaped abrasive particle of item 100, wherein Los3 isessentially the same as Los2.

Item 109. The shaped abrasive particle of item 100, wherein Los3>Lss3.

Item 110. The shaped abrasive particle of item 100, wherein Lss3>Los3.

Item 111. The shaped abrasive particle of item 100, further comprising alength factor (Los3/Lss3) of not greater than about 1, not greater thanabout 0.95, not greater than about 0.9, not greater than about 0.85, notgreater than about 0.8, not greater than about 0.75, not greater thanabout 0.7, not greater than about 0.65, not greater than about 0.6, notgreater than about 0.55, not greater than about 0.5, not greater thanabout 0.45, not greater than about 0.4, not great not greater than about0.35, not greater than about 0.3, not greater than about 0.35, notgreater than about 0.3, not greater than about 0.25, not greater thanabout 0.2, not greater than about 0.15, not greater than about 0.1, notgreater than about 0.05.

Item 112. The shaped abrasive particle of item 100, further comprising alength factor (Lss3/Los3) of not greater than about 1, not greater thanabout 0.95, not greater than about 0.9, not greater than about 0.85, notgreater than about 0.8, not greater than about 0.75, not greater thanabout 0.7, not greater than about 0.65, not greater than about 0.6, notgreater than about 0.55, not greater than about 0.5, not greater thanabout 0.45, not greater than about 0.4, not great not greater than about0.35, not greater than about 0.3, not greater than about 0.35, notgreater than about 0.3, not greater than about 0.25, not greater thanabout 0.2, not greater than about 0.15, not greater than about 0.1, notgreater than about 0.05.

Item 113. The shaped abrasive particle of item 56, wherein the bodycomprises a Shape Index of at least about 0.52, at least about 0.53, atleast about 0.54, at least about 0.55, at least about 0.56, at leastabout 0.57, at least about 0.58, at least about 0.59, at least about0.60, at least about 0.61, at least about 0.62, at least about 0.63, atleast about 0.64, at least about 0.65, at least about 0.66, at leastabout 0.67, at least about 0.68, at least about 0.69, at least about0.70, at least about 0.71, at least about 0.72, at least about 0.73, atleast about 0.74, at least about 0.75, at least about 0.76, at leastabout 0.77, at least about 0.78, at least about 0.79, at least about0.80, at least about 0.81, at least about 0.82, at least about 0.83, atleast about 0.84, at least about 0.85, at least about 0.86, at leastabout 0.87, at least about 0.88, at least about 0.89, at least about0.90, at least about 0.91, at least about 0.92, at least about 0.93, atleast about 0.94, at least about 0.95.

Item 114. The shaped abrasive particle of item 56, wherein the bodycomprises a Shape Index of not greater than about 0.98, not greater thanabout 0.97, not greater than about 0.96, not greater than about 0.95,not greater than about 0.94, not greater than about 0.93, not greaterthan about 0.92, not greater than about 0.91, not greater than about0.90, not greater than about 0.89, not greater than about 0.88, notgreater than about 0.87, not greater than about 0.86, not greater thanabout 0.85, not greater than about 0.84, not greater than about 0.83,not greater than about 0.82, not greater than about 0.81, not greaterthan about 0.80, not greater than about 0.79, not greater than about0.78, not greater than about 0.77, not greater than about 0.76, notgreater than about 0.75, not greater than about 0.74, not greater thanabout 0.73, not greater than about 0.72, not greater than about 0.71,not greater than about 0.70, not greater than about 0.69, not greaterthan about 0.68, not greater than about 0.67, not greater than about0.66, not greater than about 0.65, not greater than about 0.64, notgreater than about 0.63, not greater than about 0.62, not greater thanabout 0.61, not greater than about 0.60, not greater than about 0.59,not greater than about 0.58, not greater than about 0.57, not greaterthan about 0.56, not greater than about 0.55, not greater than about0.54.

Item 115. The shaped abrasive particle of item 56, wherein the bodycomprises a sharpness-shape-strength factor (3SF) within a range betweenabout 0.7 and about 1.7.

Item 116. The shaped abrasive particle of item 56, wherein the body hasa 3SF of at least about 0.72, at least about 0.75, at least about 0.78,at least about 0.8, at least about 0.82, at least about 0.85, at leastabout 0.88, at least about 0.90, at least about 0.92, at least about0.95, at least about 0.98.

Item 117. The shaped abrasive particle of item 56, wherein the body hasa 3SF of not greater than about 1.68, not greater than about 1.65, notgreater than about 1.62, not greater than about 1.6, not greater thanabout 1.58, not greater than about 1.55, not greater than about 1.52,not greater than about 1.5, not greater than about 1.48, not greaterthan about 1.45, not greater than about 1.42, not greater than about1.4, not greater than about 1.38, not greater than about 1.35, notgreater than about 1.32, not greater than about 1.3, not greater thanabout 1.28, not greater than about 1.25, not greater than about 1.22,not greater than about 1.2, not greater than about 1.18, not greaterthan about 1.15, not greater than about 1.12, not greater than about1.1.

Item 118. The shaped abrasive particle of item 56, wherein the bodycomprises a strength of not greater than about 600 MPa and at leastabout 100 MPa.

Item 119. The shaped abrasive particle of item 56, wherein the bodycomprises a strength of not greater than about 590 MPa, not greater thanabout 580 MPa, not greater than about 570 MPa, not greater than about560 MPa, not greater than about 550 MPa, not greater than about 540 MPa,not greater than about 530 MPa, not greater than about 520 MPa, notgreater than about 510 MPa, not greater than about 500 MPa, not greaterthan about 490 MPa, not greater than about 480 MPa, not greater thanabout 470 MPa, not greater than about 460 MPa, not greater than about450 MPa, not greater than about 440 MPa, not greater than about 430 MPa,not greater than about 420 MPa, not greater than about 410 MPa, notgreater than about 400 MPa, not greater than about 390 MPa, not greaterthan about 380 MPa, not greater than about 370 MPa, not greater thanabout 360 MPa, not greater than about 350 MPa, not greater than about340 MPa, not greater than about 330 MPa, not greater than about 320 MPa,not greater than about 310 MPa, not greater than about 300 MPa, notgreater than about 290 MPa, not greater than about 280 MPa, not greaterthan about 270 MPa, not greater than about 260 MPa, not greater thanabout 250 MPa, not greater than about 240 MPa, not greater than about230 MPa, not greater than about 220 MPa, not greater than about 210 MPa,not greater than about 200 MPa.

Item 120. The shaped abrasive particle of item 56, wherein the bodycomprises a strength of at least about 110 MPa, at least about 120 MPa,at least about 130 MPa, at least about 140 MPa, at least about 150 MPa,at least about 160 MPa, at least about 170 MPa, at least about 180 MPa,at least about 190 MPa, at least about 200 MPa, at least about 210 MPa,at least about 220 MPa, at least about 230 MPa, at least about 240 MPa,at least about 250 MPa, at least about 260 MPa, at least about 270 MPa,at least about 280 MPa, at least about 290 MPa, at least about 300 MPa,at least about 310 MPa, at least about 320 MPa, at least about 330 MPa,at least about 340 MPa, at least about 350 MPa, at least about 360 MPa,at least about 370 MPa, at least about 380 MPa, at least about 390 MPa,at least about 400 MPa, at least about 410 MPa, at least about 420 MPa,at least about 430 MPa, at least about 440 MPa, at least about 450 MPa,at least about 460 MPa, at least about 470 MPa, at least about 480 MPa,at least about 490 MPa, at least about 500 MPa.

Item 121. The shaped abrasive particle of item 56, wherein the bodycomprises a tip sharpness within a range between not greater than about80 microns and at least about 1 micron.

Item 122. The shaped abrasive particle of item 56, wherein the bodycomprises a tip sharpness of not greater than about 78 microns, notgreater than about 76 microns, not greater than about 74 microns, notgreater than about 72 microns, not greater than about 70 microns, notgreater than about 68 microns, not greater than about 66 microns, notgreater than about 64 microns, not greater than about 62 microns, notgreater than about 60 microns, not greater than about 58 microns, notgreater than about 56 microns, not greater than about 54 microns, notgreater than about 52 microns, not greater than about 50 microns, notgreater than about 48 microns, not greater than about 46 microns, notgreater than about 44 microns, not greater than about 42 microns, notgreater than about 40 microns, not greater than about 38 microns, notgreater than about 36 microns, not greater than about 34 microns, notgreater than about 32 microns, not greater than about 30 microns, notgreater than about 38 microns, not greater than about 36 microns, notgreater than about 34 microns, not greater than about 32 microns, notgreater than about 30 microns, not greater than about 28 microns, notgreater than about 26 microns, not greater than about 24 microns, notgreater than about 22 microns, not greater than about 20 microns, notgreater than about 18 microns, not greater than about 16 microns, notgreater than about 14 microns, not greater than about 12 microns, notgreater than about 10 microns.

Item 123. The shaped abrasive particle of item 56, wherein the bodycomprises a tip sharpness of at least about 2 microns, at least about 4microns, at least about 6 microns, at least about 8 microns, at leastabout 10 microns, at least about 12 microns, at least about 14 microns,at least about 16 microns, at least about 18 microns, at least about 20microns, at least about 22 microns, at least about 24 microns, at leastabout 26 microns, at least about 28 microns, at least about 30 microns,at least about 32 microns, at least about 34 microns, at least about 36microns, at least about 38 microns, at least about 40 microns, at leastabout 42 microns, at least about 44 microns, at least about 46 microns,at least about 48 microns, at least about 50 microns, at least about 52microns, at least about 54 microns, at least about 56 microns, at leastabout 58 microns, at least about 60 microns, at least about 62 microns,at least about 64 microns, at least about 66 microns, at least about 68microns, at least about 70 microns.

Item 124. The shaped abrasive particle of item 56, wherein the bodycomprises an additive, wherein the additive comprises an oxide, whereinthe additive comprises a metal element, wherein the additive comprises arare-earth element, wherein the additive comprises a dopant material,wherein the dopant material includes an element selected from the groupconsisting of an alkali element, an alkaline earth element, a rare earthelement, a transition metal element, and a combination thereof, whereinthe dopant material comprises an element selected from the groupconsisting of hafnium, zirconium, niobium, tantalum, molybdenum,vanadium, lithium, sodium, potassium, magnesium, calcium, strontium,barium, scandium, yttrium, lanthanum, cesium, praseodymium, chromium,cobalt, iron, germanium, manganese, nickel, titanium, zinc, and acombination thereof.

Item 125. The shaped abrasive particle of item 124, wherein the dopantcomprises MgO, and wherein the body comprises a content of MgO of atleast about 0.5 wt %, at least about 0.6 wt %, at least about 0.7 wt %,at least about 0.8 wt %, at least about 0.9 wt %, at least about 1 wt %,at least about 1.1 wt %, at least about 1.2 wt %, at least about 1.3 wt%, at least about 1.4 wt %, at least about 1.5 wt %, at least about 1.6wt %, at least about 1.7 wt %, at least about 1.8 wt %, at least about1.9 wt %, at least about 2 wt %, at least about 2.1 wt %, at least about2.2 wt %, at least about 2.3 wt %, at least about 2.4 wt %, at leastabout 2.5 wt %.

Item 126. The shaped abrasive particle of item 124, wherein the bodycomprises a content of MgO of not greater than about 8 wt %, not greaterthan about 7 wt %, not greater than about 6 wt %, not greater than about5 wt %, not greater than about 4.9 wt %, not greater than about 4.8 wt%, not greater than about 4.7 wt %, not greater than about 4.6 wt %, notgreater than about 4.5 wt %, not greater than about 4.4 wt %, notgreater than about 4.3 wt %, not greater than about 4.2 wt %, notgreater than about 4.1 wt %, not greater than about 4 wt %, not greaterthan about 3.9 wt %, not greater than about 3.8 wt %, not greater thanabout 3.7 wt %, not greater than about 3.6 wt %, not greater than about3.5 wt %, not greater than about 3.4 wt %, not greater than about 3.3 wt%, not greater than about 3.2 wt %, not greater than about 3.1 wt %, notgreater than about 3 wt %, not greater than about 2.9 wt %, not greaterthan about 2.8 wt %, not greater than about 2.7 wt %, not greater thanabout 2.6 wt %, not greater than about 2.5 wt %.

Item 127. The shaped abrasive particle of item 56, wherein the bodycomprises at least about 95 wt % alumina for the total weight of thebody, at least about 95.1 wt %, at least about 95.2 wt %, at least about95.3 wt %, at least about 95.4 wt %, at least about 95.5 wt %, at leastabout 95.6 wt %, at least about 95.7 wt %, at least about 95.8 wt %, atleast about 95.9 wt %, at least about 96 wt %, at least about 96.1 wt %,at least about 96.2 wt %, at least about 96.3 wt %, at least about 96.4wt %, at least about 96.5 wt %, at least about 96.6 wt %, at least about96.7 wt %, at least about 96.8 wt %, at least about 96.9 wt %, at leastabout 97 wt %, at least about 97.1 wt %, at least about 97.2 wt %, atleast about 975.3 wt %, at least about 97.4 wt %, at least about 97.5 wt%.

Item 128. The shaped abrasive particle of item 56, wherein the bodycomprises not greater than about 99.5 wt % alumina for the total weightof the body, not greater than about 99.4 wt %, not greater than about99.3 wt %, not greater than about 99.2 wt %, not greater than about 99.1wt %, not greater than about 99 wt %, not greater than about 98.9 wt %,not greater than about 98.8 wt %, not greater than about 98.7 wt %, notgreater than about 98.6 wt %, not greater than about 98.5 wt %, notgreater than about 98.4 wt %, not greater than about 98.3 wt %, notgreater than about 98.2 wt %, not greater than about 98.1 wt %, notgreater than about 98 wt %, not greater than about 97.9 wt %, notgreater than about 97.8 wt %, not greater than about 97.7 wt %, notgreater than about 97.6 wt %, not greater than about 97.5 wt %.

Item 129. The shaped abrasive particle of item 56, wherein the bodyconsists essentially of alumina and MgO.

Item 130. The shaped abrasive particle of item 56, wherein the bodyconsists essentially of alumina.

Item 131. The shaped abrasive particle of item 56, wherein the bodycomprises a polycrystalline material including crystalline grains,wherein the average grain size is not greater than about 1 micron, notgreater than about 0.9 microns, not greater than about 0.8 microns, notgreater than about 0.7 microns, not greater than about 0.6 microns.

Item 132. The shaped abrasive particle of item 131, wherein the averagegrain size is at least about 0.01 microns, at least about 0.05 microns,at least about 0.06 microns, at least about 0.07 microns, at least about0.08 microns, at least about 0.09 microns, at least about 0.1 microns,at least about 0.12 microns, at least about 0.15 microns, at least about0.17 microns, at least about 0.2 microns.

Item 133. The shaped abrasive particle of item 56, wherein the body isessentially free of a binder, wherein the body is essentially free of anorganic material, wherein the body is essentially free of rare earthelements, wherein the body is essentially free of iron.

Item 134. The shaped abrasive particle of item 56, wherein the body isformed from a seeded sol gel.

Item 135. The shaped abrasive particle of item 56, wherein the bodycomprises a primary aspect ratio of width:length of at least about 1:1and not greater than about 1:10.

Item 136. The shaped abrasive particle of item 56, wherein the bodycomprises a secondary aspect ratio defined by a ratio of width:heightwithin a range between about 5:1 and about 1:1.

Item 137. The shaped abrasive particle of item 56, wherein the bodycomprises a tertiary aspect ratio defined by a ratio of length:heightwithin a range between about 6:1 and about 1:1.

Item 138. The shaped abrasive particle of item 56, wherein the bodycomprises a polycrystalline material, wherein the polycrystallinematerial comprises grains, wherein the grains are selected from thegroup of materials consisting of nitrides, oxides, carbides, borides,oxynitrides, diamond, and a combination thereof, wherein the grainscomprise an oxide selected from the group of oxides consisting ofaluminum oxide, zirconium oxide, titanium oxide, yttrium oxide, chromiumoxide, strontium oxide, silicon oxide, and a combination thereof,wherein the grains comprise alumina, wherein the grains consistessentially of alumina.

Item 139. The shaped abrasive particle of item 56, wherein the bodycomprises a two-dimensional polygonal shape as viewed in a plane definedby a length and width, wherein the body comprises a two dimensionalshape having at least 4 exterior points, at least 5 exterior points, atleast 6 exterior points, at least 7 exterior points, at least 8 exteriorpoints, at least 9 exterior points.

Item 140. The shaped abrasive particle of item 56, wherein the body iscoupled to a substrate as part of a fixed abrasive, wherein the fixedabrasive article is selected from the group consisting of a bondedabrasive article, a coated abrasive article, and a combination thereof.

Item 141. The shaped abrasive particle of item 140, wherein thesubstrate is a backing, wherein the backing comprises a woven material,wherein the backing comprises a non-woven material, wherein the backingcomprises an organic material, wherein the backing comprises a polymer,wherein the backing comprises a material selected from the groupconsisting of cloth, paper, film, fabric, fleeced fabric, vulcanizedfiber, woven material, non-woven material, webbing, polymer, resin,phenolic resin, phenolic-latex resin, epoxy resin, polyester resin, ureaformaldehyde resin, polyester, polyurethane, polypropylene, polyimides,and a combination thereof.

Item 142. The shaped abrasive particle of item 140, wherein the shapedabrasive particle is part of a plurality of a first type of shapedabrasive particles, wherein a majority of the first type of shapedabrasive particles are coupled to the backing in an open coat, whereinthe open coat comprises a coating density of not greater than about 70particles/cm2, not greater than about 65 particles/cm2, not greater thanabout 60 particles/cm2, not greater than about 55 particles/cm2, notgreater than about 50 particles/cm2, at least about 5 particles/cm2, atleast about 10 particles/cm2.

Item 143. The shaped abrasive particle of item 140, wherein the shapedabrasive particle is part of a plurality of a first type of shapedabrasive particles, wherein a majority of the first type of shapedabrasive particles are coupled to the backing in a closed coat, whereinhaving a closed coat of the blend of shaped abrasive particles on abacking, wherein the closed coat comprises a coating density of at leastabout 75 particles/cm2, at least about 80 particles/cm2, at least about85 particles/cm2, at least about 90 particles/cm2, at least about 100particles/cm2.

Item 144. The shaped abrasive particle of item 140, wherein the shapedabrasive particle is part of a blend including a plurality of a firsttype of shaped abrasive particles and a third type of abrasive particle,wherein the third type of abrasive particle comprises a shaped abrasiveparticle, wherein the third type of abrasive particle comprises adiluent type of abrasive particle, wherein the diluent type of abrasiveparticle comprises an irregular shape.

Item 145. The shaped abrasive particle of item 144, wherein the blend ofabrasive particles comprises a plurality of shaped abrasive particles,and wherein each shaped abrasive particle of the plurality of shapedabrasive particles is arranged in a controlled orientation relative to abacking, the controlled orientation including at least one of apredetermined rotational orientation, a predetermined lateralorientation, and a predetermined longitudinal orientation.

Item 146. The shaped abrasive particle of item 140, wherein the backingcomprises an additive selected from the group consisting of catalysts,coupling agents, currants, anti-static agents, suspending agents,anti-loading agents, lubricants, wetting agents, dyes, fillers,viscosity modifiers, dispersants, defoamers, and grinding agents.

Item 147. The shaped abrasive particle of item 140, further comprisingan adhesive layer overlying the backing, wherein the adhesive layercomprises a make coat, wherein the make coat overlies the backing,wherein the make coat is bonded directly to a portion of the backing,wherein the make coat comprises an organic material, wherein the makecoat comprises a polymeric material, wherein the make coat comprises amaterial selected from the group consisting of polyesters, epoxy resins,polyurethanes, polyamides, polyacrylates, polymethacrylates, poly vinylchlorides, polyethylene, polysiloxane, silicones, cellulose acetates,nitrocellulose, natural rubber, starch, shellac, and a combinationthereof.

Item 148. The shaped abrasive particle of item 147, wherein the adhesivelayer comprises a size coat, wherein the size coat overlies a portion ofthe plurality of shaped abrasive particles, wherein the size coatoverlies a make coat, wherein the size coat is bonded directly to aportion of the plurality of shaped abrasive particles, wherein the sizecoat comprises an organic material, wherein the size coat comprises apolymeric material, wherein the size coat comprises a material selectedfrom the group consisting of polyesters, epoxy resins, polyurethanes,polyamides, polyacrylates, polymethacrylates, polyvinyl chlorides,polyethylene, polysiloxane, silicones, cellulose acetates,nitrocellulose, natural rubber, starch, shellac, and a combinationthereof.

Item 149. The shaped abrasive particle of item 56, wherein the bodycomprises a length (l), a width (w), and a height (hi), wherein thelength>width, the length>height, and the width>height, and wherein theheight (h) is at least about 100 microns, at least about 110 microns, atleast about 120 microns, at least about 150 microns, at least about 175microns, at least about 200 microns, at least about 225 microns, atleast about 250 microns, at least about 275 microns, or even at leastabout 300 microns, and not greater than about 3 mm, such as not greaterthan about 2 mm, not greater than about 1.5 mm, not greater than about 1mm, or even not greater than about 800 microns, not greater than about600 microns, not greater than about 500 microns, not greater than about475 microns, not greater than about 450 microns, not greater than about425 microns, not greater than about 400 microns, not greater than about375 microns, not greater than about 350 microns, not greater than about325 microns, not greater than about 300 microns, not greater than about275 microns, or even not greater than about 250 microns. microns, atleast about 450 microns, at least about 475 microns, at least about 500microns, and not greater than about 3 mm, not greater than about 2 mm,not greater than about 1.5 mm, not greater than about 1 mm, not greaterthan about 800 microns.

Item 150. The shaped abrasive particle of item 149, wherein the width isat least about 600 microns, at least about 700 microns, at least about800 microns, at least about 900 microns, and not greater than about 4mm, not greater than about 3 mm, not greater than about 2.5 mm, notgreater than about 2 mm.

Item 151. The shaped abrasive particle of item 56, wherein the bodycomprises a percent flashing of not greater than about 40%, not greaterthan about 35%, not greater than about 30%, not greater than about 25%,not greater than about 20%, not greater than about 18%, not greater thanabout 15%, not greater than about 12%, not greater than about 10%, notgreater than about 8%, not greater than about 6%, not greater than about4%.

Item 152. The shaped abrasive particle of item 56, wherein the bodycomprises a dishing value (d) of not greater than about 2, not greaterthan about 1.9, not greater than about 1.8, not greater than about 1.7,not greater than about 1.6, not greater than about 1.5, not greater thanabout 1.2, and at least about 0.9, at least about 1.0.

Item 153. The shaped abrasive particle of item 56, wherein the shapedabrasive particle is part of a plurality of a first type of shapedabrasive particles, wherein a majority of the first type of shapedabrasive particles are coupled to a backing in a side orientation,wherein at least about 55% of the shaped abrasive particles of theplurality of shaped abrasive particles are coupled to the backing in aside orientation, at least about 60%, at least about 65%, at least about70%, at least about 75%, at least about 77%, at least about 80%, and notgreater than about 99%, not greater than about 95%, not greater thanabout 90%, not greater than about 85%.

Item 154. The shaped abrasive particle of item 56, wherein the sidesurface comprises a first side section and a first oblique side section,and wherein at least a portion of the first oblique side sectioncomprises a curvature.

Item 155. The shaped abrasive particle of item 56, wherein the sidesurface comprises a first side section and a first oblique side section,and wherein the entire length of the first oblique side sectioncomprises a curvature.

Item 156. The shaped abrasive particle of item 56, wherein the sidesurface comprises a first side section and a first oblique side section,and wherein the first oblique side section comprises a monotonic curve.

Item 157. The shaped abrasive particle of item 56, wherein the sidesurface comprises a first side section and a first oblique side section,and wherein the first oblique side section comprises a curvaturedefining an arc segment of a circle and defining a radius of the firstoblique side section (Ros1).

Item 158. The shaped abrasive particle of item 157, wherein the radiusof the first oblique side section (Ros1) is different than the length ofthe first oblique side section (Los1).

Item 159. The shaped abrasive particle of item 157, wherein the radiusof the first oblique side section (Ros1) is greater than the length ofthe first oblique side section (Los1).

Item 160. The shaped abrasive particle of item 157, wherein the radiusof the first oblique side section (Ros1) is different than the length ofthe first side section (Lss1).

Item 161. The shaped abrasive particle of item 157, wherein the radiusof the first oblique side section (Ros1) is greater than the length ofthe first side section (Lss1).

Item 162. The shaped abrasive particle of item 157, wherein the radiusof the first oblique side section (Ros1) is less than the length of thefirst side section (Lss1).

Item 163. The shaped abrasive particle of item 157, wherein the radiusof the first oblique side section (Ros1) is less than a total length ofthe first side section (Lss1) and the first oblique side section (Los1).

Item 164. The shaped abrasive particle of item 154, wherein the firstside section and the first oblique side section are joined at a firstoblique angle (Ao1) having an obtuse value.

Item 165. The shaped abrasive particle of item 164, wherein the sidesurface further comprises a second side section and a second obliqueside section, wherein the second side section is coupled to the firstoblique section at a first external corner, wherein the first externalcorner defines a first external corner angle (Aec1) and wherein thefirst external corner angle is different than a value of the firstoblique angle (Ao1).

Item 166. The shaped abrasive particle of item 165, wherein at least aportion of the second oblique side section comprises a curvature.

Item 167. The shaped abrasive particle of item 165, wherein the entirelength of the second oblique side section comprises a curvature.

Item 168. The shaped abrasive particle of item 165, wherein the secondoblique side section comprises a monotonic curve.

Item 169. The shaped abrasive particle of item 165, wherein the firstexternal corner is defined by a joint between a curved portion of thefirst oblique side section and a linear portion of the second sidesection.

Item 170. The shaped abrasive particle of item 165, wherein the secondoblique side section comprises a curvature defining an arc segment of acircle and defining a radius of the second oblique side section (Ros2).

Item 171. The shaped abrasive particle of item 165, wherein the firstoblique side section comprises a curvature defining an arc segment of acircle and defining a radius of the first oblique side section (Ros1),wherein Ros1 and Ros2 are substantially the same.

Item 172. The shaped abrasive particle of item 165, wherein the firstoblique side section comprises a curvature defining an arc segment of acircle and defining a radius of the first oblique side section (Ros1),wherein Ros1 and Ros2 are different compared to each other.

Item 173. The shaped abrasive particle of item 56, wherein the bodycomprises a central axis and the body is asymmetric about the centralaxis.

Item 174. The shaped abrasive particle of item 56, wherein the bodycomprises at least three different central axes and the body isasymmetric about each of the three different central axes.

Item 175. The shaped abrasive particle of item 56, wherein the sidesurface of the body comprises at least 5 different side sections.

Item 176. The shaped abrasive particle of item 56, wherein the sidesurface of the body comprises at least two linear portions separated byat least one curved portion including a first side section and a secondside section separated from each other by at least a first oblique sidesection.

Item 177. The shaped abrasive particle of item 56, wherein the sidesurface of the body comprises linear portions and curved portions whichare alternating with respect to each other.

Item 178. The shaped abrasive particle of item 56, wherein the sidesurface of the body comprises linear portions and curved portions andwherein each linear portion is joined to at least one curved portion.

Item 179. The shaped abrasive particle of item 56, wherein the sidesurface of the body comprises linear portions and curved portions andwherein each linear portion is joined to at least one curved portion atan exterior corner.

EXAMPLES Example 1

Four samples of shaped abrasive particles were tested for comparison ofperformance. A first sample, Sample S1, was initially formed from amixture including approximately 45-50 wt % boehmite. The boehmite wasobtained from Sasol Corp. as Catapal B and modified by autoclaving a 30%by weight mixture of the Catapal B with deionized water and nitric acid.The nitric acid-to-boehmite ratio was approximately 0.025 in theautoclave and treated at 100° C. to 250° C. for a time ranging from 5minutes to 24 hours. The autoclaved Catapal B sol was then dried byconventional means. One may also use an alternative boehmite,commercially available as Disperal from Sasol Corp. The boehmite wasmixed and seeded with 1% alpha alumina seeds relative to the totalalumina content of the mixture. The alpha alumina seeds were made bymilling of corundum using conventional techniques, described for examplein U.S. Pat. No. 4,623,364. The mixture also included 45-50 wt % waterand 2.5-7 wt % additional nitric acid depending on the desired viscosityof the mixture, which were used to form the gel mixture. The ingredientswere mixed in a planetary mixer of conventional design and mixed underreduced pressure to remove gaseous elements from the mixture (e.g.,bubbles).

After gelling, the mixture was deposited by hand into openings of aproduction tool made of stainless steel. The openings in the productiontool were open to both sides of the production tool, such that they wereapertures extending through the entire thickness of the production tool.The cavities or openings of the production tool had a shapeapproximately the same as the shape of the particles provided herein.All samples were made with a production tool made of stainless steelwith the exception of the particles of Sample S7, which was made with aproduction tool made of PEEK. The surfaces of the openings in theproduction tool were coated with a lubricant of olive oil to facilitateremoval of the precursor shaped abrasive particles from the productiontool. The gel was placed in the openings of the production tool anddried at room temperature for at least 12 hours. After drying, theprecursor shaped abrasive particles were removed from the screen andsintered between 1250-1400° C. for approximately 10 minutes.

The shaped abrasive particles of Sample S1 had a two-dimensional shapeof an equilateral triangle as provided in the image of FIG. 27, havingan average width of about 1400 microns and height of approximately 300microns. The body was formed essentially of a seeded sol-gel aluminamaterial having an average grain size of less than 1 micron. The shapedabrasive particles of Sample S1 had an average strength of approximately847 MPa, an average tip sharpness of approximately 20 microns, a ShapeIndex of approximately 0.5, and a 3SF of approximately 1.7.

A second sample, Sample S2, was formed using the same process used toform the shaped abrasive particles of Sample S1. Sample S2 includedshaped abrasive particles having a two-dimensional, pentagon shape asprovided in the image of FIG. 28. The body had an average width ofapproximately 925 microns and height of approximately 300 microns. Thebody was formed essentially of a seeded sol-gel alumina material havingan average grain size of less than 1 micron. The shaped abrasiveparticles of Sample S2 had an average strength of approximately 847 MPa,an average tip sharpness of approximately 20 microns, a Shape Index ofapproximately 0.5, and a 3SF of approximately 1.7.

A third sample, Sample S3, was formed using the same process used toform the shaped abrasive particles of Sample S1. Sample S3 includesshaped abrasive particles having an oblique, truncated two-dimensionalshape as provided in the image of FIG. 29. The body was formedessentially of a seeded sol-gel alumina material having an average grainsize of less than 1 micron. The body had an average width ofapproximately 925 microns and a height of approximately 300 microns. Theshaped abrasive particles of Sample S3 had an average strength ofapproximately 847 MPa, an average tip sharpness of approximately 20microns, a Shape Index of approximately 0.63, and a 3SF of approximately2.7.

A fourth sample, Sample CS4, was a conventional shaped abrasive particlecommercially available as 3M984F from 3M Corporation. The body had anaverage width of 1400 microns and a height of approximately 300 microns.The shaped abrasive particles of Sample CS4 had a rare-earth elementdoped alpha-alumina composition, an average tip sharpness ofapproximately 20 microns, an average strength of approximately 606 MPa,a Shape Index of 0.5, and a 3SF of approximately 1.2. FIG. 30 includesan image of a shaped abrasive particle from Sample CS4.

A fifth sample, Sample S5, was formed using the same process used toform the shaped abrasive particles of Sample S1. Sample S5 includedshaped abrasive particles having a two-dimensional, pentagon shape asprovided in the image of FIG. 28. The body had an average length ofapproximately 1400 microns, a width of approximately 900 microns andheight of approximately 300 microns. The body was formed of a 5 wt % MgOdoped, seeded sol-gel alumina material having an average grain size ofless than 1 micron. The shaped abrasive particles of Sample S5 had anaverage strength of approximately 557 MPa, an average tip sharpness ofapproximately 20 microns, a Shape Index of approximately 0.81, and a 3SFof approximately 2.9.

A sixth sample, Sample S6, was formed using the same process used toform the shaped abrasive particles of Sample S1. Sample S6 includesshaped abrasive particles having an oblique, truncated two-dimensionalshape as provided in the image of FIG. 29. The body was formed of a 5 wt% MgO doped, seeded sol-gel alumina material having an average grainsize of less than 1 micron. The body had an average length ofapproximately 1400 microns, an average width of approximately 900microns as measured from the longest sides and excluding the shortersides of the body, and a height of approximately 300 microns. The shapedabrasive particles of Sample S6 had an average strength of approximately557 MPa, an average tip sharpness of approximately 20 microns, a ShapeIndex of approximately 0.63, and a 3SF of approximately 1.8.

A seventh sample, Sample S7, was formed using the same process used toform the shaped abrasive particles of Sample S1, except the productiontool was made of PEEK. Sample S7 includes shaped abrasive particleshaving an oblique, truncated two-dimensional shape with curved, obliqueside sections as provided in the image of FIG. 31. The body was formedof a seeded sol-gel alumina material having an average grain size ofless than 1 micron. The body had an average length (Lmiddle) ofapproximately 1590 microns, an average width of approximately 1570microns, and a height of approximately 280 microns. The shaped abrasiveparticles of Sample S7 had an average strength of approximately 847 MPabased on the expected microstructure, an average tip sharpness ofapproximately 20 microns, a Shape Index of approximately 0.61, and a 3SFof approximately 2.5.

All samples were tested according to a single grit grinding test (SGGT)in a major surface orientation and side orientation. In conducting theSGGT, one single shaped abrasive particle is held in a grit holder by abonding material of epoxy. The shaped abrasive particle is secured inthe desired orientation (i.e., major surface orientation or side surfaceorientation) and moved across a workpiece of 304 stainless steel for ascratch length of 8 inches using a wheel speed of 22 m/s and an initialscratch depth of 30 microns. The shaped abrasive particle produces agroove in the workpiece having a cross-sectional area (AR). For eachsample set, each shaped abrasive particle completes 15 passes across the8 inch length, 10 individual particles are tested for each of theorientation and the results are analyzed. The test measures thetangential force exerted by the grit on the workpiece, in the directionthat is parallel to the surface of the workpiece and the direction ofthe groove, and the net change in the cross-sectional area of the groovefrom beginning to the end of the scratch length is measured to determinethe shaped abrasive particle wear. The net change in the cross-sectionalarea of the groove for each pass can be measured. For the SGGT, the netcross-sectional area of the groove is defined as the difference betweenthe cross-sectional area of the groove below the surface and the crosssectional area of the material displaced above the surface. Performance(Ft/A) is defined as the ratio of the tangential force to the netcross-sectional area of the groove.

The SGGT is conducted using two different orientations of the shapedabrasive particles relative to the workpiece. The SGGT is conducted witha first sample set of shaped abrasive particles in a major surfaceorientation (i.e., “front” in FIG. 18), wherein a major surface of eachshaped abrasive particle is oriented perpendicular to the grindingdirection such that the major surface initiates grinding on theworkpiece. The results of the SGGT using the sample set of shapedabrasive particles in a major surface orientation allows for measurementof the grinding efficiency of the shaped abrasive particles in a majorsurface orientation.

The SGGT is also conducted with a second sample set of shaped abrasiveparticles in a side surface orientation (i.e., “side” in FIG. 18),wherein a side surface of each shaped abrasive particle is orientedperpendicular to the grinding direction such that the side surfaceinitiates grinding of the workpiece. The results of the SGGT test usingthe sample set of shaped abrasive particles in a side orientation allowsfor measurement of the grinding efficiency of the shaped abrasiveparticles in a side orientation. Note that Sample S7 was tested in twodifferent side orientations. In the first side orientation (“Side-C”),the curved, oblique side section of the side surface defined the leadingedge of the corner during the grinding test. In the second sideorientation (“Side-S”), a linear surface section of the side surface wasthe leading edge of the external corner during the grinding test.

FIG. 32 includes a plot of median force per total area removed from theworkpiece for shaped abrasive particles tested in the front (i.e., majorsurface orientation) and side orientations (i.e., side surfaceorientation), which is representative of data derived from the SGGT forall of the samples. The force per total area removed is a measure of thegrinding efficiency of the shaped abrasive particles, with a lower forceper total area removed indicating more efficient grinding performance.As illustrated, Sample S3 demonstrated the best performance of allsamples tested in the front orientation. Such results are quiteremarkable and surprising, and without wishing to be tied to aparticular theory, it is noted that one or more combination of features,including the oblique, truncated polygonal shape, the strength, the tipsharpness, the Shape Index, and the height of Sample S3 facilitatedimproved performance over the other grains. Notably, Sample S3demonstrated approximately a 50% improvement in grinding efficiency overthe shaped abrasive particles of Samples CS4 and S2 in the frontorientation. Moreover, the Samples S3 demonstrated a 77% increase ingrinding efficiency compared to Sample S1. Moreover, Sample S7demonstrated the best overall performance in all different orientations.

Example 2

Abrasive particles having the shape and microstructure as outlined inSample S7 were formed using a machine including a die to extrude the gelmixture into openings of a production tool being translated under thedie. These grains were used to form coated abrasive samples having theconstruction outlined below and designated CAS7. Moreover, grains fromSample CS4 were sourced and formed into a coated abrasive having thesame construction as CAS7 and were designated CACS4. To form SamplesCAS7 and CACS4, a backing of finished cloth of approximately 47 poundsper ream was obtained and coated with a make formulation including aphenol formaldehyde resin as provided in Table 4. Using an electrostaticdeposition process, 41 pounds per ream of abrasive particles having theshape and microstructure as outlined for Sample S7 or CS4 was applied tothe backing with the make coat. The structure was dried in an oven fortwo hours at 80° C. It will be appreciated that the make coat is createdsuch that sum of the components provided in Table 4 equals 100%.

TABLE 4 Make Coat Formulation Make Formulation Component PercentageFiller NYAD Wollastonite 400 45-50 wt % Wet Witcona 1260 0.10-.2 wt %Resin, SI 45-50 wt % Solmod Silane A1100 0.1-3 wt % Water 0.1-1 wt %

The coated abrasive structure was then coated with a size coat havingthe formulation presented in Table 5. The construction was heat treatedin an oven set for a final soak temperature of 100-120° C., in which thesample was held for approximately 20-30 minutes. It will be appreciatedthat the size coat is created such that sum of the components providedin Table 5 equals 100%.

TABLE 5 Size Coat Formulation Size Formulation Component Percentage Dye2-4 wt % Solmod Tamol 165A 0.5-2 wt % Filler Syn Cryolite K 40-45 wt %Resin Single Comp 94-908 50-55 wt % DF70 Defoamer 0.1-0.2 wt % Water 2-4wt %

The coated abrasive sample was then placed into an oven to undergo heattreatment. The oven temperature was set for a final soak temperature ofapproximately 110-120° C., in which the sample was held forapproximately 10-12 hours.

A supersize coat having the formulation provided below in Table 6 wasthen applied to Samples CAS7 and CACS4 and processed in the same manneras the size coat. It will be appreciated that the supersize coat iscreated such that sum of the components provided in Table 6 equals 100%.

TABLE 6 Supersize Coat Formulation Supersize Formulation ComponentPercentage Dye 1-3 wt % Solmod Cabosil 0.05-3 wt % Solmod DAXAD 11 1-4wt % Filler Type A 63-67 wt % Resin PF Prefere 80-5080A 20-25 wt % DF70Defoamer 0.1-0.2 wt % Water 6-10 wt %

Each of the coated abrasive samples CACS4 and CAS7 was tested accordingto a standardized grinding test using the conditions summarized in Table7. Two samples of coated abrasives were tested in each case to derivethe results.

TABLE 7 Test conditions: Test mode: Dry, straight plunge Constant MRR′ =4 inch³/min/inch Belt speed (Vs) = 7500 sfpm (38 m/s) Work material: 304ss Hardness: 96-104 HRB Size: 0.5 × 0.5 × 12 inches Contact width: 0.5in Contact Wheel: Steel Measurements: Power, Grinding Forces, MRR′ andSGE Cum MR compared at SGE = 2.4 Hp · min/inch³

FIG. 33 includes a plot of specific grinding energy versus cumulativematerial removed (at a material removal rate of 4 inch³/min inch) foreach of the samples. It is notable and quite remarkable that Sample CAS7had a significantly lower specific grinding energy during the majorityof the test compared to Sample CACS4 despite being a blockier grain witha higher Shape Index compared to the grains of Sample CACS4. Thus,without wishing to be tied to a particular theory, it appears that theperformance of a shaped abrasive particle and the associated fixedabrasive in which it is incorporated can be achieved by controlling thecombination of sharpness, strength, and shape of the abrasive particles.

The present application represents a departure from the state of theart. Conventional shaped abrasive particles have previously focused onmaking triangular shaped grains having the sharpest possible corners andedges. However, through empirical studies of shaped abrasive particleshaving various shapes and microstructures, it has been discovered thatcertain grain features (e.g., tip sharpness, strength, and Shape Index)appear to be interrelated and may be controlled with respect to eachother to provide improved performance of a shaped abrasive particle.Additionally, as noted herein, the height may be related as well.Notably, in the present application, it is noted that one may notnecessarily need to create a shaped abrasive particle with the sharpestfeatures, but instead may control one or more of a combination of grainfeatures, including tip sharpness, strength, Shape Index, and heightrelative to each other to improve the grinding performance of a shapedabrasive particle beyond conventional shaped abrasive particles. Inparticular, it is noted that the Shape Index may define an overall shapeof the body and how stress is distributed throughout the body duringgrinding, which when combined with a suitable tip sharpness andstrength, may provide improved results over conventional triangularshaped abrasive particles having sharp tips. Moreover, while notcompletely understood and without wishing to be tied to a particulartheory, it is thought that one or a combination of these features of theembodiments described herein facilitate the remarkable and unexpectedperformance of these particles in fixed abrasives, such as coatedabrasive and bonded abrasives.

Certain features, for clarity, described herein in the context ofseparate embodiments, may also be provided in combination in a singleembodiment. Conversely, various features that are, for brevity,described in the context of a single embodiment, may also be providedseparately or in any subcombination. Further, reference to values statedin ranges includes each and every value within that range.

Benefits, other advantages, and solutions to problems have beendescribed above with regard to specific embodiments. However, thebenefits, advantages, solutions to problems, and any feature(s) that maycause any benefit, advantage, or solution to occur or become morepronounced are not to be construed as a critical, required, or essentialfeature of any or all the claims.

The specification and illustrations of the embodiments described hereinare intended to provide a general understanding of the structure of thevarious embodiments. The specification and illustrations are notintended to serve as an exhaustive and comprehensive description of allof the elements and features of apparatus and systems that use thestructures or methods described herein. Separate embodiments may also beprovided in combination in a single embodiment, and conversely, variousfeatures that are, for brevity, described in the context of a singleembodiment, may also be provided separately or in any subcombination.Further, reference to values stated in ranges includes each and everyvalue within that range. Many other embodiments may be apparent toskilled artisans only after reading this specification. Otherembodiments may be used and derived from the disclosure, such that astructural substitution, logical substitution, or another change may bemade without departing from the scope of the disclosure. Accordingly,the disclosure is to be regarded as illustrative rather thanrestrictive.

The description in combination with the figures is provided to assist inunderstanding the teachings disclosed herein. The following discussionwill focus on specific implementations and embodiments of the teachings.This focus is provided to assist in describing the teachings and shouldnot be interpreted as a limitation on the scope or applicability of theteachings. However, other teachings can certainly be used in thisapplication.

As used herein, the terms “comprises,” “comprising,” “includes,”“including,” “has,” “having” or any other variation thereof, areintended to cover a non-exclusive inclusion. For example, a method,article, or apparatus that comprises a list of features is notnecessarily limited only to those features but may include otherfeatures not expressly listed or inherent to such method, article, orapparatus. Further, unless expressly stated to the contrary, “or” refersto an inclusive-or and not to an exclusive-or. For example, a conditionA or B is satisfied by any one of the following: A is true (or present)and B is false (or not present), A is false (or not present) and B istrue (or present), and both A and B are true (or present).

Also, the use of “a” or “an” is employed to describe elements andcomponents described herein. This is done merely for convenience and togive a general sense of the scope of the invention. This descriptionshould be read to include one or at least one and the singular alsoincludes the plural, or vice versa, unless it is clear that it is meantotherwise. For example, when a single item is described herein, morethan one item may be used in place of a single item. Similarly, wheremore than one item is described herein, a single item may be substitutedfor that more than one item.

Unless otherwise defined, all technical and scientific terms used hereinhave the same meaning as commonly understood by one of ordinary skill inthe art to which this invention belongs. The materials, methods, andexamples are illustrative only and not intended to be limiting. To theextent not described herein, many details regarding specific materialsand processing acts are conventional and may be found in reference booksand other sources within the structural arts and correspondingmanufacturing arts.

The above-disclosed subject matter is to be considered illustrative, andnot restrictive, and the appended claims are intended to cover all suchmodifications, enhancements, and other embodiments, which fall withinthe true scope of the present invention. Thus, to the maximum extentallowed by law, the scope of the present invention is to be determinedby the broadest permissible interpretation of the following claims andtheir equivalents, and shall not be restricted or limited by theforegoing detailed description.

The Abstract of the Disclosure is provided to comply with Patent Law andis submitted with the understanding that it will not be used tointerpret or limit the scope or meaning of the claims. In addition, inthe foregoing Detailed Description of the Drawings, various features maybe grouped together or described in a single embodiment for the purposeof streamlining the disclosure. This disclosure is not to be interpretedas reflecting an intention that the claimed embodiments require morefeatures than are expressly recited in each claim. Rather, as thefollowing claims reflect, inventive subject matter may be directed toless than all features of any of the disclosed embodiments. Thus, thefollowing claims are incorporated into the Detailed Description of theDrawings, with each claim standing on its own as defining separatelyclaimed subject matter.

What is claimed is:
 1. A shaped abrasive particle comprising: a bodycomprising a first major surface, a second major surface, a first sidesurface extending between the first major surface and the second majorsurface, and a second side surface extending between the first majorsurface and the second major surface, wherein the first side surfacecomprises a first side section and a first oblique section, wherein thefirst side section and the first oblique section are joined at a firstoblique corner, wherein the first oblique corner has a first obliquecorner angle, wherein the first oblique corner angle has a value of atleast 92 degrees and not greater than 176 degrees, wherein the firstside surface and the second side surface come together at a firstexternal corner, wherein the first external corner has at first externalcorner angle, wherein the first external corner angle has a value lessthan the value of the first oblique corner angle, and wherein the bodycomprises alumina, and further wherein the body comprises asharpness-shape-strength factor (3SF) of at least 1.3, a strength of atleast 100 MPa, a Shape Index of at least 0.51 and not greater than 0.80,and wherein the body comprises a tip sharpness of at least 2 microns andnot greater than 50 microns.
 2. The shaped abrasive particle of claim 1,wherein the first side section has a first side section length and thefirst oblique section has a first oblique section length, and whereinthe first side section length is different than the first obliquesection length.
 3. The shaped abrasive particle of claim 2, wherein thefirst side surface has a length factor of at least 0.05 and not greaterthan 0.95, wherein the length factor is the first oblique section lengthdivided by the first side section length.
 4. The shaped abrasiveparticle of claim 1, wherein a ratio of the value of the first externalcorner angle to the value of the first oblique corner angle is at least0.05 and not greater than 0.95.
 5. The shaped abrasive particle of claim1, wherein the body comprises a tip sharpness within a range between notgreater than about 40 microns and at least about 1 micron.
 6. The shapedabrasive particle of claim 1, wherein the body comprises an additivecomprising dopant material selected from the group consisting of analkali element, an alkaline earth element, a rare earth element, atransition metal element, and a combination thereof.
 7. The shapedabrasive particle of claim 1, wherein the body comprises apolycrystalline material including crystalline grains, wherein theaverage grain size is not greater than 1 micron.
 8. The shaped abrasiveparticle of claim 1, wherein the first oblique section has a concavecurvature.
 9. The shaped abrasive particle of claim 1, wherein the bodyis coupled to a substrate as part of a fixed abrasive selected from thegroup consisting of a bonded abrasive article, a coated abrasivearticle, and a combination thereof.
 10. The shaped abrasive particle ofclaim 1, wherein the body comprises at least 95 wt % a lumina.
 11. Theshaped abrasive particle of claim 1, wherein the body comprises astrength of at least 300 MPa.
 12. The shaped abrasive particle of claim1, wherein the body comprises at least four external corners.
 13. Theshaped abrasive particle of claim 1, wherein the body comprises at leastfour distinct side surfaces.
 14. The shaped abrasive particle of claim1, wherein the body comprises a shape index of not greater than 0.69.15. The shaped abrasive particle of claim 1, wherein the body isessentially free of an organic material.
 16. The shaped abrasiveparticle of claim 1, wherein the body is essentially free of rare earthelements.
 17. The shaped abrasive particle of claim 1, wherein the bodyis essentially free of iron.
 18. The shaped abrasive particle of claim1, wherein the body is formed from a seeded sol gel.